an elementary course in the integral calculus
TRANSCRIPT
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AN ELEMENTARY COURSE
IN
INTEGRAL CALCULUS
BT
DANIEL ALEXANDER MURRAY, Ph.D.
INSTRUCTOR IN MATHEMATICS IN CORNELL UNIVERSITY
FORMERLY SCHOLAR AND FELLOW OF JOHNS HOPKINS UNIYBRSITY
AUTHOR OF"
INTRODUCTORY COURSE IN
DIFFERENTIAL EQUATIONS''
"o"9{o"-
NEW YORK.;. CINCINNATI.:. CHICAGO
AMERICAN BOOK COMPANY
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'/; r."":""."
' ^ i'
(viMV 21 189-;
I.
"^^ R A ?^--^'^
''r-y---
COPTRIOHT, 1898, BT
AMERICAN BOOK COMPANY.
JlirK"AY*8 INTBO. OAUX
W. P. 2
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PREFACE
This book has been written primarily for use at Cornell
University and similar institutions. In this university the
classes in calculus are composed mainly of students in engi-eering
for whom an elementary course in the Integral Calculus
is prescribed for the third term of the first year. Their pur-ose
in taking the course is to acquire facility in performing
easy integrations and the power of making the simple applica-ions
which arise in practical work. While the requirements
of this special class of students have been kept in mind, care
has also been taken to make the book suitable for any one be-innin
the study of this branch of mathematics. The volume
contains little more than can be mastered by a student of
average ability in a few months, and an effort has been made
to present the subject-matter,hichis of an elementary char-cter,
in a simple manner.
Theobjectof the first two chapters is to give the student
a clear idea of what the Integral Calculus is, and of the uses
to which it may be applied. As this introduction is somewhat
longer than is usual in elementary works on the calculus, some
teachers may, perhaps, prefer to postpone the reading of sev-eral
of the articles until the student has had a certain amoxmt
of practice in.the processes of integration. It is believed, how-ver,
that a careful study of Chapters I.,II., will arouse the stu-ent's
interest and quicken his understanding of the subject.There may be some difference of opinion also as to whether
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VI PREFACE
the beginner should be introduced to the subjectthrough
Chapter I. or through Chapter II. The decision of this ques-ion
will depend upon the point of view of the individual teacher.
So far as the remaining portion of the book is concerned it is
a matter of indifference which of these chapters is taken first;
and, with slight modifications, they can be interchanged. In
Chapter III. the fundamental rules and methods of integration
are explained. Since it has been deemed advisable to intro-uce
practical applications as early as possible, Chapter IV. is
devoted to the determination of plane areas and of volumes of
solids of revolution. Thesubjectf
Integral Curves, which is of
especial importance to the engineer, is treated in Chapter XII.
Many of the examples are original. Others, especially some
of those given in the practical applications, by reason of their
nature and importance, are common to all elementary courses
on calculus. In several instances, examples of particular interest
have been drawn from other works.
A list of lessons suggested for a short course of eleven or
twelve weeks is given on page viii. This list has been arranged
so that four lessons and a review will be a week's work.
It is hardly possible to name all the sources from which the
writer of an elementary work may have obtained suggestions
and ideas. I am especially conscious, however, of my indebted-ess
to the treatises of De Morgan, Williamson, Edwards,
Stegemann and Kiepert, and Lamb.
To my colleagues in the department of mathematics at
Cornell University, I am under obligations for many valuable
criticisms and suggestions. Both the arrangement and the
contents have been influenced in a large measure by our con-ferences
and discussions. As originally projected,he volume
was to have been written in collaboration with Dr. Hutchinson,
but circumstances prevented the carrying-out of this plan.
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PREFACE vii
Chapters V., VI., in part, and Articles 2S, 73, in their entirety,
have been contributed by him. My colleagues have aided me
also in correcting the proofs.
Erom Professor I. P. Church of the College of Civil Engi-eering
and Professor W. Y, Durand of the Sibley College of
Mechanical Engineering, I have received valuable suggestions
for making the book useful to engineering students. Pro-essor
Durand kindly placed at my disposal, with other notes,
his article on ^'Integral Curves" in the Sibley Journal of
Engineering, Vol. XI., No. 4 ; and Chapter XII. is,with slight
changes, a reproduction of that article. I take this oppor-unity
of thanking Mr. A. T. Bruegel, Instructor in the kine-atics
of machinery, and Mr. Murray Macneill, Fellow in
mathematics in this university, the former for the interest
and care taken by him in drawing the figures, the latter for
his assistance in verifying examples and reading proof sheets.
D. A. MURRAY.
Cornell Univbbsitt.
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LIST OF LESSONS SUGGESTED FOR A SHORT COURSE
[Rbvikws to follow Every Foueth Lesson]
10.
IL
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Arts. 1, 2, 3.
Arts. 4, 5, 6, 7.
Arts. 8, 9, 10, 11.
Arts. 12, 17, 18, to Ex. 9.
Arts. 13, 18, Exs. 10-12, 19.
Arts. 14, 20.
Arts. 15, 16, 21.
Arts. 22, 23, Exs. 1-23, odd
examples.
Art. 23, Exs. 24-32, 24, 25,
Exs. 1-8, page 55.
Pages 56, 57, even or odd exam-ples,
Exs. 9-41, Exs. 42-47.
Arts. 26, 27, Exs. 1, 2, page 68.
Arts. 28, 29, Exs. 3, 4, page 68.
Exs. 5-14, page 68.
Art. 30.
Arts. 31, 32.
Page 76, Exs. 1-15.
Pages 76, 77, Exs. 16-26.
Arts. 33, 34.
Arts. 35, 36. Selected examples.
Arts. 37, 38, 39, 40.
Arts. 42, 43.
Art. 45.
Selected examples, pages 98, 99.
Arts. 46, 47, 48 (a).
Arts. 48 (rf),9, 50 (a),(c).Art. 51 (a).
Arts. 52, 53.-
Arts. 58, 59, 60.
Arts. 61, 62..
Arts. 64, 65, 66.
Arts.67, 68, 69.
Arts. 63, 70. Selected exam-ples,
pages 164, 165.
Art. 71. Selected examples,
page 165.
Art. 72. Selected examples,
page 166.
Art. 73.
Art. 74. Selected examples,
pages 164, 165.
Art. 75, Ex. 9, page 164.
Art. 76. Selected examples.
Art. 77, Ex. 1.
Selected examples, pages 162-
166.
Arts. 78, 79. Selected examples.
Art. 80. Selected examples.
Arts. 81, 82, 84, 85.
Arts. 86, 87, 88.
Arts. 89, 91, 92, 94, 95.
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CONTENTS
CHAPTER I
Intbobation a Process of Summation
Abt. Page
1. Uses of the integral calculus. Definition and sign of integration.
1
2. Illustrations of the summation of infinitesimals....
2
3. Geometrical principle .7
4. Fundamental theorem. Definite integral 8
6. Supplement to Art. 3 13
6. Geometrical representation of an integral 14
7. Properties of definite integrals 16
CHAPTER II
Integration THE Inverse op Differentiation
8. Integration the inverse of differentiation 18
0. Indefinite integral. Constant of integration 22
10. Geometrical meaning of the arbitrary constant of integration.
23
11. Relation between the indefinite and the definite integral. .
24
12. Examples that involve anti-differentials 25
13. Another derivation of the integration formula for an area. .
27
14. A new meaning of y in the curve whose equation is y =/(x).De-ived
curves 29
15. Integral curves 33
16. Summary.35
CHAPTER ra
Fundamental Rules and Methods of Integration
18. Fundamental integrals '
. .
.3619. Two universal formulae of integration 39
20. Integration aided by a change of the independent variable . .41
21. Integration by parts"
. 44
22. Additional standard forms 47
23. Derivation of the additional standard forms. . . . ,
48
ix
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X CONTENTS
Abt. Paok
24. Integration of a total differential 52
25. Summary 54
Examples on Chapter 111.55
CHAPTER IV
Geometrical Applications of the Calculus
26. Applications of the calculus . . . ... . .
.58
27. Areas of curves, rectangular coordinates 58
28. Precautions to be taken in finding areas by integration...
63
29. Precautions to be taken in evaluating definite integrals...
67
30. Volumes of solids of revolution 69
31. On the graphical representation of a definite integral...
73
32. Derivation of the equations of certain curves . .
*
. .
.74
Examples on Chapter IV 76
CHAPTER V
Rational Fractions
34. Case I 79
35. Case II 81
36. Casern 82
CHAPTER VI
Irrational Functions
38. The reciprocal substitution 84
39. Trigonometric substitutions 85
40. Expressions containing fractional powers of a-\-
hx only . .85
41. Functions of the form / " x^, (a + 6"^)"
\'Xdx,'mwhich w, 7i, are
integers 87
42. Functions of the form F(x, Vx^ + ax + 6) dx, F(u, v)being a
rational function of u,v 88
43. Functions of the form /(.r,V" x^-f
ax + 6)dx, f{u, v)being a
rational function of u, v 89
44. Particular functions involving Vax^ + dx-f-
c 91
45. Integration of x"'(a+ hx'^ydx : (a) by the method of undetermined
coefficients ; (6)by means of reduction formulae.
-
.
.93Examples on Chapter VI 98
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CONTENTS xi
CHAPTER Vn
Iktboration of Trigonometric and Exponential Functions
Abt. Page
46. rsiu^xdic,cos^xdx,being an integer 100
47. Algebraic transformations 103
48. fsec*a;"fe,
fcosec"xdx 103
49. ftan"X(te,fcot"xdx 106
60. fsin""xcos*X(te 107
51. Integration of sin""x cos" dx: (a) by the method of undetermined .
coefficients; (6) by means of reduction formulae. .
109
62. rtan""x8ec"x(ix,jcot""xcosec*xdx 113
63. Use of multiple angles.114
54. f ^115
J a^co8^z-\- ft^sin^x
66. f ^ , f f 115Ja + bcosx Ja + bsmx
66. ie^sinnxdx, Je^cosnxdx 117
57. I sin mx cos nx (2x, (cosmxcosux(2x, I sin 7"x sin nx(Zx . .118
CHAPTER Vin
Successive Integration. Multiple Integrals
68. Successive integration 119
69. Successive integration with respect to a single independent variable 119
60. Successive integration with respect to two or more independent
variables . . . 123
61. Application of successive integration to the measurement of areas:
rectangular coordinates 126
62. Applicationof
successive integration to the measurement
ofvolumes : rectangular coordinates 128
63. Further application of successive integration to the measurement
of volumes : polar coordinates 131
CHAPTER IX
Further Geometrical Applications. Mean Values
65. Derivation of the equations of curves in polar coordinates . . 134
66. Areas of curves when polar coordinates are used : by single inte-ration
135
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Xll CONTENTS
Art. Vaqm
67. Areas of curves when polar coordinates are used: by doable
integration 139
68. Areas in Cartesian co5rdinates with oblique axes.... 140
69. Integration after a change of variable. Integration when the vari-bles
are expressed in terms of another variable .
,
.141
70. Measurement of the volumes of solids by means of infinitelythin
cross-sections 142
71. Lengths of curves : rectangular coordinates 144
72. Lengths of curves : polar co5rdinates 147
73. The intrinsic equation of a curve 149
74. Areas of surfaces of solids of revolution 152
75. Areas of surfaces whose equations have the formz=f(z,y) .
156
76. Mean values 160
77. A more general definition of mean value 163
Examples on Chapter IX 164
CHAPTER X
Applications to Mechanics
78. Mass and density 167
79. Center of mass 168
80. Moment of inertia. Radius of gyration 173
CHAPTER XI
Approximate Integration. Integration bt Means of Series. Integra-ion
BY Means of the Measurement op Areas
81. Approximate integration 177
82. Integration in series 177
83. Expansion of functions by means of integration in series .
.17984. Evaluation of definite integrals by the measurement of areas J81
85. The trapezoidal rule 182
86. The parabolic, or Simpson's one-third rule 184
87. Durand's nile 187
88. The planimeter 188
CHAPTER Xn
Integral Curves
89. Introduction 190
90. Special case of differentiation under the sign of integration. . 190
91. Integral curves defined. Their analytical relations . . . 192
92. Simple geometrical relations of integral curves.... 194
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CONTENTS xiii
An. Paob
03. Simple mechanical relations and applications of integral curves.
Successive moments of an area about a line....
105
04. Practical determination of an integral curve from itsfundamental
curve. The integraph 198
0" The determination of scales 200
CHAPTER Xin.
Obbiiiabt Differential Equations
06. Differential equation. Order. Degree 201
07. Constants of integration. General and particular solutions. Deri-ation
of a differential equation 202
Section L Equationsof the First Order and the First Degree
08. Equations in which the variables are easily separable . . .206
00. Equations homogeneous in x and y ..... .205
100. Exact differential equations .206
101. Equations made exact by means of integrating factors. .
.207
102. Linear equations 208
103. Equations reducible to the linear form 200
Section IL Equationsof the First Order but not of the First Degree
104. Equations that can be resolved into component equations of the
firstdegree 210
105^ Equations solvable for y 211
106. Equations solvable for x 212
107. Clairaut's equation .213108. Geometrical applications. Orthogonal trajectories . .
214
Section III. Equations of an Order Higher than the First
100. Equations of the form ^= /(") 218
dx^
110. Equations of the form ^= /(y) 218
111. Equations in which y appears in only two derivatives whose orders
differ by unity 210
112. Equations of the second order with one variable absent 220
113. Linear equations. General properties. Complementary functions.
Particular integral 222114. The linear equation with constant coefficients and second member
zero 223
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XIV CONTENTS
Art. Paqi
115. Case of the auxiliary equation having equal roots . . .224
116. The homogeneous linear equation with the second member zero .225
Examples on Chapter XIII 227
APPENDIX
Note A. A method of decomposing a rational fi*action into its partial
fractions 229
Note B. To find reduction formulae for Jaf^^Cahx^y dx by integra-ion
by parts 231
Note C. To find reduction formulae for fsin""a; cos" x dx by integra-ion
by parts 233
Note D. A theorem in the infinitesimal calculus 234
Note E. Further rules for the approximate determination of areas.
235
Note F. The fundamental theory of the planimeter . . . .237
Note G. On integral curves 240
1. Applications to mechanics 240
2. Applications in engineering and in electricity . .242
3. The theory of the integraph 244
Figures of some of the curves referred to in the examples . . .240
A short table of integrals 249
Answers to the examples .263
Index 285
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INTEGRAL CALCULUS
CHAPTER I
INTEGRATION A PROCESS OF SUMMATION
1. Uses of the integral calculus. Definition and sign of integration.
The integral:calpulus can be used for two purposes, namely:
(a) To find the sum of an infinitely large number of infinitesi-als
of the form f(x)dx ;
(b) To find the function whose differential or whose differential
coefficient is given ; that is,to find an anti-differentialor an anti-
derivative.
The integral calculus was invented in the course of an en-deavor
to calculate the plane area bounded by curves. The area
was supposed to be divided into an infinitely great number of
infinitesimal parts, each part being called an element of the area ;
and the sum of these parts was the area required. The process
of finding this sum was called integration, a name which implies
the combination of the small areas iutd a whole, and hence the
sum itself was called the whole or the integral
From the point of view of the firstof the purposes justindi-ated,
integration may be defined as a process of stimmation. In
many of the applications of the integral calculus, and, in particu-ar,
in the larger number of those made by engineers, this is the
definition to be taken. On the other hand, however, in many
problems it is not a sum, but merely an anti-differential,that is
required. For this purpose, integration may be defined as an
operation which is the inverseofdifferentiation.t may at once be
1
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2 INTEGRAL CALCULUS [Ch. I.
stated that in the course of making a summation by means of the
integral calculus it will be necessary to find the anti-differential
of some function ; and it may also be said at this point, that the
anti-differentialcan be shown to be the result of making a sum-mation.
Each of the above definitions of integration can be de-ived
from the other. These statements will be found verified in
Arts. 4, 11, 13.
In the differentialcalculus, the letter d is used as the symbol
of differentiation,and df(x)is read "the differentialoif(x)," In
the integral calculus the symbol of integration is*
|, andif(x)x
is read "the integraloif(x)dx,'^
The signs d and I are signs of
operations ; but they also indicate the results of the operations
of differentiation and integration respectively on the functions
that are
written afterthem.
The principal aims of this book are : (1)to explain how sum-mations
of infinitesimals of the form f(x)dx may be made ; (2)to
show how the anti-differentialsof some particular functions may
be obtained.
2. Illustrations of the summation of infinitesimals. Two simple
illustrations of the summation of an infinitenumber of infinitely
small quantities will now be given. They will help to familiarize
the student with a certain geometrical principle and with the
fundamental theorem of the integral calculus, which are set forth
in Arts. 3, 4. The method employed in these particular instances
is identical with that used in the general case which follows them.
* This is merely the long Sj which was used as a sign of summation by
the earlier writers, and meant** the sum of." The sign Jwas first employed
in 1675, and is due to Gottfried Wilhelm Leibniz (1646-1716),ho invented
the differential calculus independently of Newton. The word integral ap-eared
first in a solution of James Bernoulli (1654-1705),which was pub-ished
in the Acta Ernditorum, Leipzig, in 1690. Leibniz had called the
integral
calculus calculus summatoriu$,
but in 1696 the term calculus in-
tegralis was agreed upon between Leibniz and John Bernoulli (1667-1748).
See Cajori,History of Mathematics^ pp. 221, 237.
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1-2.] INTEGRATION A PROCESS OF SUMMATION
(a) Find the area between the line whose equation is y = mz,
the a?-axis,and the ordinates for which a? = a, a: = 6.
Let OL be the line y = mx', let OA be equal to a, and OB to 6,
and draw the ordinates APy BQ. It is required to find the area
of APQB, Divide the segment AB into n parts, each equal to
L
Y
Aa?; and at the points of section ^" A^y """, erect ordinates AiP^,
AiP^ ..., which meet OL in Pi, P^ .... Through P, Pi, Pg, ..., Q,
draw lines parallel to the axis of x and intersecting the nearest
ordinate on each side, as shown in Fig. 1, and produce PBi to
meet BQ in C
It will firstbe shown that the area APQB is the limit of the
sum of the areas of the rectangles P^i, Pi^2" """" when n, the
number of equal divisions of AB, approaches infinity, or, what
is the same thing, when Aa? approaches zero. The area APQB
is greater than the sum of the "inner^^
rectangles P^i, Pi^2" """ ;
and itis less than the sum of the "outer "rectangles ^Pi, AiP^, """.
The difference between the sum of the inner rectangles and the
sum of the outer rectangles is equal to the sum of the small rec-tangles
PPi, Pi Pg, ."".
The latter sum is equal to
^iPiAa; + J5,P2Aa?4-- +J5"QAaj;
that is,to {B^Pi+ J^aPj 4- """ + B^Q) Aa?, which is CQ Aa?.
INTEGRAL CALC. " 2
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4"
INTEGRAL CALCULUS [Ch. I.
This may be briefly expressed,
S^Px-
SP^i = SPP,
= CQ Aa;.
When Aa; is an infinitesimal, the second member of this equa-ion
is also an infinitesimal of the firstorder ; therefore, when Aa;
isinfinitelysmall the limit of the difference between the total areas
of the inner and of the outer rectangles is zero. The area APQB
lies between the total area of all of the inner and the total area
of all of the outer rectangles. Hence, the area APQB is the
limit both of the sum of the inner rectangles and of the sum of
the outer rectangles as Aa? approaches zero. Each elementary
rectangle has the area y Aa?, that is mx Aa;, since y = mx. The
altitudes of the successive inner rectangles, going from A towards
B, are ma, m (a-|-a;), (a-f2 Aa;),"
", m (a-h(w" 1)Aa;).Hence,
Area APQB = limit^3.^o^{a Aa; + (a+ Aa;)a; -f(a+2 Aa;)a; + " " "
+ (a+ n-lAa;)Aa;{*
=
limit^^fpn\a+(a+ Ax)'i-(af 2Aa;)-f""
+ (a+ w " 1 Aa;)Aa;.
Addition of the arithmetic series in brackets gives
Area APQB =
limit^^^o^^^2a
-f(n- 1)Aa;}
= limit^^^o^" "~^^{6-f a " Aa;},ince n Aa;= 6" a,
="(f-i)" The symbol Aa;=0 means
"
when Ax approaches zero as a limit." It is
due to the late Professor Oliver of Cornell University.
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2.] INTEGRATION A PROCESS OF SUMMATION
In this example the element of area is obtained by taking a
rectangle of altitude y, that is,ma?, and width Ax, and then letting
Aoj become infinitesimal.
The expression n = oo may be used instead of Aa? = 0, since
Aaj =
h " a
It may be noted in passing, that ifthe anti-differentialof rnxdac,
namely^'^j be taken, and h and a be substituted in turn for x, the
difference between the resulting values will be the expression
obtained above.
(6)Find the area between the parabola y"^, the aj-axis,and
the ordinates for which x= a, x= b.
Let QiOQ be the parabola y = x*; let OA be equal to a,
antj
OB
to b. Draw the ordinates AP, BQ. It is required to find the
area APQB, Divide the segment AB into n parts each equal to
Fig. 2.
Aa?, and at the points of division A^ A2, """, erect ordinates AiP^,
A2P2, """. Through P, Pi, P2, """, draw lines parallel to the axis
, of X and intersecting the nearest ordinates on each side, as in
Fig. 2. It can beshown,
inthe
same
wayas in
the previousillus'
tration, that the area APQB is equal to the limit of the sum of
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6 INTEGRAL CALCULUS [Ch.I.
the rectangles P-4" FiA^, """, Pn-i-B, when Aoj approaches zero.
This area will now be calculated.
The area of any elementary rectangle is yAa?, that is o^Aa;,
since y = aJ* ; and the altitudes of the successive rectangles, going
from A toward B, are a*, (a+ ^xf,(a-f2 Aa?)*,"". Hence,
Area APQB = limit^^o{"*^a;-h(a
4- Aaj)*Aa?-|-(a+2Aa?)*Aa?f-..
+ (a4-w " lAaj)2Aa;|
= limit^^o{a'+(a Aaj)"-fa+ 2 Aaj)*
+ (a-|-n~lAaj)2|Aa?
= limit^^ol^'+ 2 a Aaj(l+2-h3H-...+w-l)
+ (Aaj)Xl'H-2"3*+ ... +ir^*)}Aa?.
It is shown in algebra that the sum of the squares of the first
n natural numbers, 1^ 2^ 3^ ..., n\is ^(^ + l)(2tt l\ r^^^
6
application of this result to the sum of squares in the second
member of the last equation and the summation of the arith-etical
series
1, 2, 3, """
(n"
1),gives
Area APQB = limit^^o^ ^x"a^-\- an^x " a Ax
+
(Aa,)'("-l)(But nAa? = 6 " a;
and hence, a* + an Aa;-f-
" (Aa;)*^(a*-f-a6 -f 6*).
Hence,^IPQB
= limit^^o("-a)i""*+^+"'
" ^^
3 3
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2-3.] INTEGRATION A PROCESS OF SUMMATION
In this example, the element of area is obtained by taking a
rectangle whose area is a^Aic and then letting Ax be an infinitesi-al.
Here also, it may be noted in passing, that if the anti-
differential of a^dx, namely "
,be taken, and b and a substituted
o
in turn for ", the difference between the resulting values will be
the expression justderived for the area.
3. Geometrical principle. Let f(x) be a continuous function
of X, and let FQ be an arc of the curve whose equation is
Fio. 8.
y=f(x).Draw the ordinates AP, BQ, and suppose OA = a,
OB = b. It is required to find the area APQB ; that is, the
area between the curve, the oj-axis,and the ordinates AP, BQ,
Divide the segment AB into n parts, each equal to Ax, and at
the points of section ^j, ^2?"""" erect the ordinates A^Pi,
-^jPj,"""
to meet the curve in P^ Pg, "... Through P, Pi, Pg, """, draw
lines parallel to the cr-axis and intersecting the nearest ordinate
on each side, as in Fig. 3, and produce PBi to meet BQ in (7.
It will nT)w be shown that the area APQB is the limit of the
sum of the areas of the rectangles PAi, P1A2, """, when the num-
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8 INTEGRAL CALCULUS [Ch. I.
ber of equal divisions of AB is made infinitely great, that is,when
Ax is made infinitely small. The area APQB is greater than the
sum of the areas of the inner rectangles PAi, PiA^, """, and less
than the sum of the areas of the outer rectangles APi, AiP^, """.
That is,
^PA,"APQB"^P,A,
The difference between the sum of the outer rectangles and the
sum of the inner rectangles is equal to the sum of the small
rectangles PPi, P1P3, """ ; that is,
= BiPi^x + J^jPjAa? 4- " + B^Q Aoj
= (APi + ^2^2 + - + B,Q)Ax
=zCQAx.
The difference, CQ Ax, can be made as small as one pleases by
decreasing Ax, Therefore, since the area of APQB always lies
between the areas of the inner and outer series of rectangles,
and since the difference between these areas approaches zero as
its limit, the area APQB is the limit both of the sum of the
inner rectangles, and of the sum of the outer rectangles.
Therefore, the area induded between the curve whose equation is
y =f(x),the Qc-axis,and a pair of ordinates, is the limit
of the sum
of the areas of the rectangles whose bases are successive segments of
the part of the x-axis intercepted by the pair of ordinates, and whose
altitudes are the ordinates erected at the points ofdivision
of the
X-axis, as the bases approach zero,
4. Fundamental theorem. Definite integral. Since the equa-ion
of the curve, an arc of which is given in Fig. 3, isy=f(x),
the heights of the successive inner rectangles, going towards the
right from A, are
/(a),fia + Ax),f(a + 2Ax),"',f{a+(n-l)Ax).
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3-4.] INTEGRATION A PROCESS OF SUMMATION 9
Hence,
Area APQB = limitAoj^o{/(a)a?
-f
/(a + Aa?)a:
+/(a + 2 Aaj)oj + """ +f(a + w - 1 Aoj)Aoj}. (1)
The second member of (1)is the limit of the sum of the
values, infinite in number, that f(x)^x takes as x varies by
equal increments Aoj from x = a to x = b, when Aa; is made
infinitesimal. This limit may be indicated by
Limits
iO/^/(")A"5*
In the integral calculus this is more briefly indicated by
prefixing to f(x)dx the sign j, at the bottom and top of which
are respectively written the values ^of x at which the summa-tion
begins and ends ; thus :
x = h
Cf(x)x.
x=a
An abbreviation for this form is
i'^f(ix)dx.
(2)
This is read, "the integral of f(x)dx between the limits
a and 6." The initial and final values of x, namely, a and 6,
are called the lower and upper limits respectively of the inte-ral.*
The differential f(x) dx is called an element of the
integral. It evidently represents the area of any one of the
component infinitesimal rectangles of altitude f{x) and infini-esimal
base dx. In the same way that dx is a differential of
* This manner of indicating the limits between which the summation
is to bemade
bywriting
the lower limitat
the bottom
andthe
upperlimit at the top of the integration sign, is due to Joseph Fourier
(1768-1830).
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10 INTEGRAL CALCULUS [Ch. I.
the distance along the aj-axis,so f{x)dx, or y dx, as it is usually
written, is a differential of the area between the curve and
the axis of x.
The limit of the sum in the second member of (1),which is
indicated by the symbol (2),will now be obtained. Suppose
that f(x)dx is d"f"(x),hat is, suppose that 4"{x)is the antir
differentialf f(x)dx. Then, by the fundamental principle of
the differential calculus,
in which e is a quantity that varies with x and approaches
zero when Aa; approaches zero. On clearing of fractions and
transposing, the latter equation becomes
/(a?)a?= "^(a? Aa;) "l"(x)e Aa?. (3)
On substituting in (3) the values of x at the successive
points of division between A and B at intervals equal to Aa?,
the following equations are obtained:
/(a)Aa? =
" (a+ Aa?) ^ (a) Cq Aa?,
/(a -f Aa?)a; =
" (a-f2 Aa?) " (a-f A")
" ^i Aa?,
/(a -f2 Aa?)a;= "^(a-f
3 Aa?) "^(a-f2 Aa?) ejAa?,
f(b " Aa?)
a? =
" (6) " (6 Aa?) e^_i Aa?,
inwhich each of the e's approaches zero when Aa? approaches
zero. The sum of the first members of these equations is
equal to the sum of the second members; that is,
y^f(x)^x"l"(b)"^(a) (eo ci + - + c"_i)^a?-
x = a
Of the quantities Cq, ei, """e"_i, suppose that e^ has an absolute
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4.] INTEGRATION A PROCESS OF SUMMATION 11
value E not less than that of any one of the others. It E ^
substituted for each of the e's, the term
becomes nE^x, or (6"
a)Ej
since n^x^h " a. Hence, ''
^/(aj)Aa:="^(6)-"^(a).
a quantity which is not greater
than (6"
a)E, and which ap-roaches
zero when E ap-roaches
zero, that is, when
Ax approaches zero.
Therefore, on letting Aa? approach zero, there will be obtained,
X /(aj)"laj +(6)- + (a).
Hence, the mm or irUegrcU, I f(x)dx, which is the 8um of aU the
vcUueSy infiniten number, thatf(x)dx takes as x varies by infinites-
imal increments from a to b, is found by obtaining the anti-differntial
"l"(x)off(x)dx, and subtra^ing the value of ^ (x)for x = a
from its value for a? = 6. The following notation is used to indi-ate
these operations :
J^/(x)(te[^(a;)J'=^(6)-4)
"It will be shown in Art. 9, that if dp(x) =f(x)dx, the anti-differential
of /(x) (te is 0 (") + c, in which c is an arbitrary constant. Hence, equation
(4)should be written
y^/(x)(to[0(a;)+c]*.
Since the same c is used when a and b are substituted for x, this becomes
JyCx)"te0(6) -0(a).as above.
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12 INTEGRAL CALCULUS [Ch. I.
The sum | f(x)dx is called a definitentegral because it has a
definite value, the limit of the series in the second member of
(1). Since the evaluation of this definite integral is equivalent
to the measurement of the area between the curve y = /(a?),he
ic-axis,and the ordinates at a; = a, a; = 6, the area may be regarded
as representing the integral. It follows from the result (4)that
a definite integral may be regarded as either :
(1)The limit of the sum of an infinitely large number of infini-esimal
quantities of the form f{x)dx taken between certain
limits; or,
(2)The difference of the values of the anti-differential of
f(xydx at each of these limits.
If f(x)is any continuous function of a?,f(x)dx has an anti-differ-ntial.*
However, the deduction of the anti-differential is often
impossible, and in any case, is less simple and easy than the
process of differentiation.f
Many of the practical applications of the integral calculus,
such as finding areas, lengths of curves, volumes and surfaces of
solids, centers of gravity, moments of inertia, mass, weight, etc.,
consist in making summations of infinitely small quantities. The
integral calculus adds these infinitesimal quantities together
and gives the result. It has been observed that in order to
obtain the sum of infinitesimal areas, etc., the anti-differential
of some function is required. Accordingly, a considerable part of
any book on the integral calculus is devoted to the exposition of
methods for obtaining the anti-differentials of functions which
frequently appear in the process of solving practical problems.
* The truth pf this statement, for all the ordinary functions, will appear
in the sequel. A proof applicable to all forms of continuous functions is
given in Picard, Traite d*Analyse^t. I., No. 4.
t The phrase "to find the anti-differential" means to deduce a. flnite
expression for the anti-differentialin terms of the well-known mathematical
functions. In cases in
whichthe
anti-differentialcannot be thus
obtained,an
approximate value of the definite integral can be found by the methods dis-ussed
in Arts. 84-88. A short inspection of these articlesmay be made now.
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4-5] INTEGRATION A PROCESS OF SUMMATION 18
5. Supplement to Art. 3. In proving the principle of Art. 3,
the arc PQ in Fig. 3 was used. If the arc of the given curve
has the form and position in Fig. 4, the proof of the principle
is as set forth in Art. 3. If the arc has the form and position in
T
Fig. 6.
Fig. 5, and thus has maximum and minimum values of the
ordinate, the principle stillholds. This can be seen by drawing
the maximum and minimum ordinates that come between AF
and BQ, and remarking that the principle holds for the several
successive parts APPiAi, AiPiPiAy """.
Suppose that the curve has the form and position in Fig. 6.
The area of the part APAi is the limit of the sum of the ele-entary
areas f(x)^x when Ax approaches zero and x varies
from OA to OAi ; or, in other words, the^area
of APAi is the
limit of the sum of the elementary areas f(x)dx as x varies from
OA to OAi. Similarly, the area of A1TA2 is the limit of the sum
of the elementary areas f(x)dx ss x varies from OAi to OA^f and
the area A^QB is the limit of the sum of the elementary areas
f(x)dx ss X varies from OA2 to OB. In APAi and A2QB, the
ordinates that represent the values of /(a?)re positive, while in
A1TA2 the ordinates are negative. Since x is taken as varying
from left to right, dx is always positive. Accordingly, areas such
as APAi, AiQB, which lie above the aj-axis,have a positive sign,
and areas such as AiTA^ which lie below the avaxis, have a neg-tive
sign. This example shows that in the case of a curve that
crosses the avaxis, the method of summation by means of the
integral calculus gives the algebraic sum of the areas that lie
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14 INTEGRAL CALCULUS [Ch. I.
between the curve and the aj-axis, the areas above the a?-axis
being given a positive sign, and those below receiving a nega-ive
sign.
If the total absolute area between the curve and the axis of
X is required, the portions APjX\, A^TA^, A^QB, should be found
separately.
Note. If n is a constant not equal to " 1, the anti-differential of u*du is
^; for, differentiation of the latter gives u^du,
n
+1
Ex. 1. Find the arfea between the curve whose equation is y = x', the
X-axis and the ordinates for which x = 1, x = 4.
By Art 4, the area required = Tx^dx
= 63| units of area.
Ex. 2. Find the area between the parabola 2y = 6x^, the x-axis and the
ordinates for which x = 2, x = 6. Ana, 97^ square units.
Ex. 8. Find the area between the line y = 4 x, the x-axis and the ordi-ates
for
whichx = 2, x = 11. An8, 2*34
square units.
Ex. 4. Find the area between the parabola 2 y = 3 x^, the x-axis and the
ordinates for which x = " 3, x = 5. Ans. 76 square units.
Ex. 5. Find the area between the line y = 5 x, the x-axis and the ordinate
for which x = 2. Ans. 10 square units.
Ex. 6. Find the area between the line y = 6x, thex-axiis and the ordinates
for which x = " 2, x = 2. Ans, 0.
6. Geometrical representation of an integral. It is necessary to
perceive clearly that a definite integral, whether it be the sum
of an infinite number of infinitesimal elements of area, length,
volume, surface, mass, force, work, etc., can be represented graphi-ally
by an area. For instance, in order to represent the definite
integral
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5-7.] INTEGRATION A PROCESS OF SUMMATION 15
choose a pair of rectangular axes, plot the curve whose equation
and draw the ordinates for x = a, x=: h. It has been shown in
Art. 4 that the area between this curve, the a?-axis,and these
ordinates has the value of the definite integral above. Hence,
this area can represent the integral. This does not mean that
the area is equal to the integral, for the integral may be a length,
a volume, etc. The area can be taken to represent the integral,
because the number that indicates the area is equal to the number
that indicates the value of the integral. That an integral may
be represented geometrically by an area is at the foundation
of some important theorems and applications of the integral
calculus.
7. Properties of definite integrals. In Art. 4 it was shown that if
the definite integral, jf(x)dx= if b) if^a).
From this, the firstof the following properties is immediately
deducible. The second and third properties depend upon Art. 6.
(a)"f(x)dx^--jy(x)dx.
This relation holds since the second member is " } (a) " (6)|;
that is,^(6)" 4"{a),Hence, the algebraic sign of a definite inte-ral
is changed by an interchange of the limits of integration.
%/a *ya %/c
Let the curve whose equation isy=f{x)
be drawn; and
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16 INTEGRAL CALCULUS [Ch.I.
let ordinates AP, BQ, CR be erected at the points for which
x= a, x=by 0? = c. Since
area APQB = area APBC
+ area CBQB,
rf(x)dx cy(x)dx+Cf{x)dx,%/a "/a "/"
It does not matter whether c is
-Xbetween a and h or not. For, sup-ose
OC = c', and draw the ordi-ate
OW ; then
C B C
Fig. 7.
area APCIB = area APE'C - area BQR'C ;
f(x)dx= I f(x)dx" I f(x)dx,
which, by (a),
=J^(aj)cte
+j[(a?)a?.
Therefore a given definite integral may be broken up into any
number of similar definite integrals that differ only in the limits
between which integration is to be performed.
(c)Construct APQB as in (6)to represent the definite integral
rf(x)dx.Then
f f(x)dx = area APQB
Q
r= area
of
a
rectangle whoseheight CB is greater
than AP and less than
BQ, and whose base
isABy
= AB'CB
=
(6-a)/(c),
OC being equal to c.
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7.] INTEGRATION A PROCESS OF SUMMATION 17
Therefore /(c)= ^ "-
0"
a
The function /(c)is called the mean value of f{x)for values of
X that vary continuously from a to 6. This mean value may be
defined to be equal to the height of a rectangle which has a base
equal to 6 " a and an area that is equivalent to the value of the
integral. Thesubjectof mean values is discussed further in
Arts. 76, 77.
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CHAPTER II
INTEGRATION THE INVERSE OF DIFFERENTIATION
8. Integration the inyerse of differentiation. In Art. 1, two
definitions of integration were indicated, namely :
(a)Integration is a process of summation ;
(b)Integration is an operation which is the inverse of differen-iation.
The first definition was discussed in Chapter I. In this and
the next following article,integration will be considered from the
point of view of the second definition.
The differential calculus is in part concerned with finding the
differential or the derivative of a given function. On the other
hand, the integral calculus is in part concerned with finding the
function when its differential or its derivative is given. If a
function be given, the differential calculus affords a means of
deducing the rate of increase of the function per unit increase of
the independent variable. If this rate of increase of a function
be known, the integral calculus affords a means of finding the
function.
Ex. 1. A curve whose equation is y = 4 ac^ isgiven ; and the rate of increase
of the ordinate per unit increase of the abscissa is required.
Since y = 4 x^,
^= 8x.
dx
This means that the ordinate at a point whose abscissa is x is increasing 8 z
times as fast as the abscissa. If this rate of increase remains uniform, the
ordinate will receive an increase of 8 x when the abscissa is increased by
unity. This determines the direction of the curve at the point.
18
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8.] INTEGRATION INVERSE OF DIFFERENTIATION 19
On the other hand, suppose that at any point on a curve, it is known that
dx
and let the equation of the curve be required. It evidently follows from this
equation that
in which c is an arbitrary constant. The constant c can receive any one of
an infinite number of values ; and hence, the number of curves that satisfy
the given condition is infinite. If an additional condition be imposed, for
example, that the curve pass through the point (2,3), then c will have a
definite corresponding value. For, since the point (2,3) is on the required
curve,
3 = 4.22 + c,
and accordingly, c = " 13.
Hence, the equation of the curve that satisfiesboth of the conditions above
given is
y = 4xa-13.
Ex. 2. In the case of a body falling from rest under the action of gravity,
the distance s through which itfallsin t seconds is a constant, approximately
16 times ^ feet ; find the velocity at any instant.
Here, " = 16f2,
andhence, ^ = S2t;
that is,the velocity* in feet per second at the end of t seconds is 32 1.
On the other hand, suppose it is known that in the case of a body falling
from rest, the velocity in feet per second is 32 multiplied by the time in
seconds since motion began. Let the corresponding relation between the
distance and the time be required.
daHere, it is known that " = 32 ".
'
dt
" If a body moves in a straight line through a distance As in a time M, and
if its average velocity be denoted by v,
As
At
As At approaches zero. As also approaches zero, and the velocity approaches
dathe definite limiting value ^"
dt
INTEGRAL CALC. " 3
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20 INTEGRAL CALCULUS [Ch. II.
It is obvious that the solution of this simple differential equation is
(1) "=:16"2 + C,
inwhich
c is an
arbitrary constant.Tliis
resultis indefinite. By
the condi-ionsof the question, however, " = 0 when i = 0.
Hence, substituting in(l), 0 = 0 + c;
whence, c = 0,
and " = 16"2
is the definite solution.
The distance fhroughwhich
the body falls can
alsobe deter-ined
by the method of summation employed in the firstchapter.
Let the number 32 be denoted by g. The distance passed over in
any time is equal to the product of the average velocity duiing
that time and the time. The time t may be divided into n equal
intervals A^, so that ^ = nAf. The velocity at the beginning of
therth
interval is
(r 1)Af,
and atthe
end ofthe interval is
rg A^. Hence, the distance passed over in the interval liesbetween
(r-l)g{My^XLdLrg{M)\
On finding similar limits for the distance passed over in the
case of each of the intervals and adding, it will be found that the
total distance passed over lies between
[0+ 1 + 2 + - + (n - V)-\g{Myand [1+ 2 + - + n^gi/Hty-,
that is, summing these arithmetical series, the distance passed
over lies between
Since A^ =-, the distance lies betweenn
2 2n 2^2n'
and the distance is the common limit of these two expressions,
when A^ approaches zero, that is, when n approaches infinity.
Hence, 8 = ^gt^,
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8.] INTEGRATION INVERSE OF DIFFERENTIATION 21
Sometimes the anti-differential(orthe anti-derivative)f a func-ion
is wanted for its own sake alone, as in the illustrations
given above ; and sometimes it is desired for further ends, as, for
example, in the process of making a summation by means of
the integral calculus in Art. 4. The anti-differentialis called the
integraly the process of finding it is called integration, and the
symbol of integration is the sign I.
Thus, if the differential of
" (x)is/(")dx, which is expressed by
d"^(aj)=/(aj)daj, (1)
then Cf(x)dx"fi(x),
" (2)
Equation (1)may be read" the differential of " (x)
isf(x)dx ;"
equation (2)may be read" the function of which the differential
is f{x)dx, is"fi")."
For brevity, the latter may be read"the
integral of f(x)dxis "^(aj).""
Memory of the fundamental formulae of differentiation will
carry one far in the integral calculus. For instance, since rfa^
is 4a^(22r, I4tx^dx is ic*; since dsina; is cos a; da:, |cos a:da? is
sin X, The beginner will see the necessity of having ready com-mand
of the formulae for differentiation, since they will be
employed in the inverse process of integration.t Differentiation
* The origin of the terms integral, integration, and of the sign i has
been given in Art. 1. Instead of the sign 1,the symbols d~^ and 2"-i are
sometimes employed:thus,
(?-y(x)(to,hich is read **the
anti-differentialof f(x)dx,
' '
and D-^f{x) which is read,* *the anti-derivative of f(x)/
' In the
case of the second definition of integration, the use of the symbols d~\ D~^,
is more logical than the use of \. The latter sign is, however, fii-mly estab-ished
in this connection. It may be remarked that the differential is more
frequently written than is the derivative of a fimction.
t The expressions Xx^dx, d''\x^dx),D'^ixP) are equivalent. The in-erse
process of
integration is notalways practically possible (see
Art.
4).Art. 81 may be referred to for examples of differentials whose integrals can-ot
be expressed in a finiteform.
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22 INTEGRAL CALCULUS [Ch. IJ.
of both members of (2)gives*
whence, by (1),=/(^)
^^"
Therefore, d neutralizes the effect of 1.
It will be shown in
the next articlethat jd"^(aj)ay have values different from" (").
9. Indefinite integral. Constant of integration. Since d(Qi^-{-c)
is 4aj'da?, c being any constant, l4:a^dxis a^ + c. The integral
given in Art. 8 comes from this on making c zero. But c may
be given any other value that does not involve x. Hence, i 4ta^dx
is indefinite so far as an arbitrary added constant is concerned.
In general,
if d"tt(x)=f(x)dx, (1)
then ff(x)dx=if"{x)c, (2)
in which c is any constant ; for differentiation of the members of
(2)shows that f{x)dx=idit"(x).ence, the integral of a given
differentials indefiniteo far as an arbitrary added constant is
concerned. Illustrations have been seen in the preceding articles.
It should be noted that the indefiniteness does not extend to
terms that contain x. In other words, a given differential can
have an infinitenumber of integrals that correspond to the infi-ite
number of valuesthat an
arbitrary constantcan take, but
any two of these integrals differ only by a constant. For
instance,
/'x-^l)dx= ^-\-x-\-Ci.
But on substituting zfovx-h 1, and consequently, dz for dx,
J(x-^l)dx=fzdz^^-]^c,==^^^^-hc,^
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8-10.] INTEGRATION INVERSE OF DIFFERENTIATION 23
These two integrals agree in the terms that contain x. When
an integration is performed, the arbitrary constant should be
indicated in the result ; or, if not indicated, it should be under-tood
to be there. The second member of (2)is usually called
the indefinitentegral of f(x)dx, and c is said to be the constant
of integration. When the constant of integration has an arbitrary
value, that is,when no definite value has- been assigned to it,the
integral is called also a general integral ; on the other hand, when
the constant of integration is given a particular value, the integral
is said to be a particular integral. For instance, the general
integral (and indefinite integral)f oj*da? is ^a^ + c in which
c is arbitrary. A particular integral of a^dx is obtained by
giving c any one of an infinity of possible values, say 6, " 5, ^.
Thus \3ii^6y \iii^5, ^ix^-hi are particular integrals. In
practice the value of the constant may be determined by the
special conditions of the problem.
10. Geometrical meaning of the arbitrary constant of integration.
If |=-^(*)' (1)
then y = I^\^) ^y
that is, y = F(x)+ c, (2^
in which c is an arbitrary constant of integration. Suppose that
c is
given particular values, say
8, " 3, etc. ;and
let the curves
whose equations are
y = i^(aj)+8,|
etc., etc.,
be drawn. All of these curves have the same value of -X ; thatdx
is, the same direction, for the same value of a?. Also, for any
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24 INTEGRAL CALCULUS [Ch.il
Fig. 9.
two of the curves the difference in the lengths of their ordinates
remains the same for all
values of x. For example,
for any value of x, the dif-erence
between the lengths
of the ordinates of the two
curves whose equations are
given in (3)is 8-(-3),
or 11. Hence, all the
curves, whose equations are
of the form (2),hus differ-ng
merely in the c's, can be obtained by moving any one of the
curves vertically up or down. The particular value assigned
to c merely determines the position of the curve with respect to
the oj-axis,andhas
nothingto do
withits form. Fig. 9 illustrates
this.
11. Relation between the indefinite and the definite integral. In
Art. 4 it has been seen that if d"l"x)=f(x)dx, the sum or inte-ral
of f{x)dx for all values of x from x = a to x = b, satisfies
the relation
"f(x)dx="f^(b)^"k(a)1)
If the upper limit be variable and be denoted by x,
"f(x)d^"f^(x)^4"(a). (2)
If the lower limit a be arbitrary, "
" (a)may be represented
by an arbitraiy constant c, and (2)becomes
But
Jf{x)dx"l"(x)+c.
Jf(x)dx=^if"(x)c
(3)
(4)
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10-12.] INTEOBATION INVERSE OF DIFFERENTIATION 26
Hence, an indefinite integral is an integral whose upper limit
is the variable and whose lower limit is arbitrary. The first
member of (4)may be considered an abbreviation for the first
member of (3). The indefinite integral may. therefore be re-garded
as obtained by a process of summation.
12. Examples that involve anti-differentials. This article is in-erted
for the purpose of giving simple, typical examples of a
practicalkind, in
which anti-differentials are requiredfor
pur-oses
other than that of summation. These illustrations will
also involve the determination of constants. In many applica-ions
of the calculus, two kindsof constants must be distinguished,
namely, those which are constants of integration, and those
which are given distances, angles, forces, etc. ; for example, the
constantg,
in Ex. 2, Art. 8,
and
h, 6, k, a, in Ex. 2 below. Rec-angular
coordinates are used in the following exercises.
Ex. 1. Determine the equation of the curve at every point of which the
tangent has the slope J. Determine the equation of the curve which passes
through the point (2, 3) and also satisfies the former condition. The
slope of a curve y "f(x) at any point (x,y)is
-^.*Hence, by the given
condition,
(1)^
=1
^^dx 2
Adopting the differentialform, dy = idXy
and integrating, (2) y = ^ x + c,
the equation of a straight line. Now c, the arbitrary constant of integration,
which in this case represents the intercept of the line on the j/-axis,can take
an infinite number of values. The first condition is therefore satisfied by
each and all of the parallel lines of slope ^.
If, in addition, the line is required to pass through the point (2,3),then
X = 2, y = 3, satisfy (2), and S = \ -2 -\-c. From this, c = 2. Hence, the
curve that satisfiesboth of the given conditions is the line whose equation is
y = ix-^2.
* By the slope of a curve at any point is meant the tangent of the angle
that the tangent line to the curve at the point makes with the x-axis.
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26 INTEGRAL CALCULUS [Ch. II.
Ex. 2. Determine the equation of the curve that shall have a constant
subnormal. Also, determine the curve which has a constant subnormal and
passes through the two given points (o,h), (",k); and find the length of its
constant subnormal.
Let A, B, be the given points (o,h),(6,k),and let P be any point (x,y)on the curve. Suppose that PT is the tangent at P, and PH the normal.
Put the constant subnormal 3f2V equal to a.
Since the angle a = angle 0 (seeFig. 10),their sides being respectively
perpendicular,
tan a = tan $ ;
that is, " !=-:"Putting this in the differentialform,
(2) ydy^adz.
and integrating both sides,^-^c'
= ax-\- c",
whence. (3) |- ax + c,
in which c', c", are the constants of integration, and c denotes c" " c'.
Equation (3)isthe equation of a parabola, and it includes an infinitenumber
of parabolas, one for each of the infinite number of values that the arbitrary
constant c can have.
The particular curve which passes through the points (o,h) (6,k), and
has a constant subnormal is also required. Since the coordinates of these
points must satisfy (3),it follows that
andF
= ab -\-c.
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12-13.] INTEGBATION INVERSE OF DIFFERENTIATION 27
These equations suffice to determine c and the length a of the constant
subnormal. On solving them, it is found that
c=iA2, a = -
26
Hence, the equation of the secondcurve required is
In the differential calculus it is shown that the length of the subnormal
is y
-^.The firstcondition might have been expressed immediately by the
equation
which is equivalent to (1)and (2).
Ex. 3. Find the curve whose subtangent is constant and equal to a. De-ermine
the curve so that it shall pass through the point (0,1).
Ex. 4. Find the curve for which the length of the subnormal is propor-ional
to (sayk times)the length of the abscissa.
13. Another derivation of the integration formula for an area.
Ill Arts. 3, 4, it was shown that the area included between the
curve y =/(a;),he avaxis, and the ordinates for x = a, x=h is
the limit of the sum of the infinitely large number of infinites-mal
quantities f{x)dx,which are successively obtained as x
varies continuously from a to b, and this limit was represented
by the definite integral I f(x)dx. The area can also be derived
by means of the second defi-ition
of integration.
Let CPQ be an arc of the
curve whose equation is y =
f(x),and let OA = a,OB = b.
Draw the ordinates AP, BQ,
Take any point S on the
avaxis at a distance x from
0, and draw the ordinate SL whose length is /(a?).Let z
denote the area of OCLS. If x or OS is increased by ST,
which is equal to Ax, and the ordinate TM be drawn, the area
z will be increased by the area SLMT. This increment will
Fro. 11.
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28 INTEGRAL CALCULUS [Ch. II.
be denoted by A2;. DrawXi? parallel to the ovaxis, and complete
the rectangle LRMV in which LB = "^x and EM= Ay.
The increment of the area,
i^z^SLMT
= rectangle SLRT-h
area LRM
= SLRT-\- an area less than the rectangle LRMV
=/(")Aa:
+ arealess
than Ay" Aa.
Hence,
A2"
=/(ic)-h something less than Ay.Ao;
When Aa? approaches zero, Ay also approaches zero ; and hence,
in the limit,
|=/(^); (1)
that is, J = y. (2)
Equation
(2)means
that the numerical value ofthe differential
coefficient with respect to the abscissa, of the area between a
curve, the axes of coordinates, and an ordinate, is the same as the
numerical value of this ordinate of the curve.
Equations (1)and (2)written in the differential form give the
differential of this area, namely,
dz=f{x)
dx, and dz = ydx, (3)
Finding the anti-differentialsin (3)gives as the area OGLS,
I dx
=
*(a?)+c,(4)
in which "^(a:)s the anti-derivative of f(x)and c is an arbitrary
:=J/(a!),
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13-14.] INTEGRATION INVERSE OF DIFFERENTIATION 29
constant. If the area is measured from the j^-axis,the area is
zero when x is zero. Hence, substituting these values in (4)
0 =
"^(0)-hc,
whence, c = "
"^(0),
and (4)becomes z =
"f"(x)"^(0).
Hence, area of OCPA =
"^(a) "^(0).
Similarly, area of OCQB =
"^(6) "^(0).
Since APQB = OCQB - OCPA,
it follows that area APQB =
"^(6) "t"(a).
The expression in the second member is the same that was
found for the area in Art. 4 by means of the first definition of
integration.
If the area is measured, not from the y-axis, but from another
fixed vertical line, say the ordinate at x = m, the derivation of
equations (1)and (2)is the same as given above. In this case,
the area is zero when x = m, and hence,
0 =
"^(m)c. From this, c = "
"f"(m).
The value of c in (4)thus depends solely upon the fixed ordi-ate
from which the area is measured.
14. A new
meaning of yin the curve
whose equationis
i/=
f(x).Derived curves. It has been seen in the differential calculus
that in the case of a curve whose equation isy=f(x), at any
point on the curve the slope of the curve is-^,
the differentialdx
coefficient of its ordinate with respect to its abscissa. Art. 13,
with equations (2)and (4),hows that at any point of a curve
the length of the ordinate y is the differential coefficient with
respect to the abscissa, of the area bounded by the curve, the
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80 INTEGRAL CALCULUS [Ch. II.
axis of X, a fixed ordinate, and the ordinate at the point.* There-ore,
" if we wish to make a graphic picture of any function and
its derivative, we can represent the function either by the ordi-ate
y of a curve or by its area, while its derivative will then be
represented by its slope or ordinate respectively. If we are most
interested in the function,e usually employ the former method
(inwhich the ordinate represents the function)if in its denva-
tive, the latter (inwhich the ordinate represents the derivative).
That is,we usually like to use the ordinate to represent the main
variable under consideration.''f
For instance, suppose it is necessary to represent the function
f(x). Let the curve be drawn whose equation is
y=f(x). (1)
At any point (a?,y)
on the curve, the ordinate y represents the
value of the function for the corresponding value of x ; and the
slope -2 represents the rate of change of the function compareddx
with the rate of change of the variable x. Now let the curve be
drawii whose equation is
y=f'(^), (2)
df(x)in which /'(a?)enotes '^^
^- At any point (a?,y) on this curve.
* The remaining part of this article is not necessary for the articles that
follow. However, it may be useful for the beginner to read it,because it
may help to strengthen his grasp on the fundamental principles of the
integral calculus.
t Irving Fisher, A briefintroduction to the Infinitesimalalculus designed
especially to aid in reading mathematical economics and statistics. Art. 89.
Some readers may be interested in an application of the principle quoted
above. Professor Fisher continues: "Jevons, in his Theoryof
Political
Economy, used the abscissa x to represent commodity, and the area z to
represent its total utility, so that its ordinate y represented*
marginal
utility' (i.e.he differential quotient of total utility with reference to com-modity).Anspitz
andLieben, on the
other
hand, in their Untersuchungen
Uber die Theorie des Preises, represent total utility by the ordinate and
marginal utility by the slope of their curve."
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14.] INTEGRATION INVERSE OF DIFFERENTIATION 31
the numerical measure of the ordinate is the same as that of the
slope of the first curve at the point having the same abscissa.
Hence, an ordinate at any point of the second curve represents
the rate of change of the function f(x)compared with the rate of
change of the variable x at this point. Also, the area between
the second curve, the axes, and the ordinate at (x,y)is
that is /(a')-/(0).
Hence, the area of the curve y=zf'(x)bounded as described
above plm a constant quantity /(O) can represent the function
For example, suppose that the function ispa^ + 4. That is,
f(x)=p:x?-\-^,
and f(x) = 2px,
The parabola y = ps? -\-4, and the line y = 2px are shown in
Fig. 12. At any point M on the avaxis, the ordinates MPj MQ
are drawn to these curves. The ordinate MP represents the
function for x= 0M\ and the slope at P represents the rate of
change of the function when x = OM, The ordinate MQ is
equal (numerically)o the slope at P\ and hence, it also repre-
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32 INTEGRAL CALCULUS [Ch. II.
sents the rate of change of the function when x = OM. More-ver,
since
area OQM = | 2px dx = pa^,
the function f(x)for x=OM is represented by the area 0Q3f +4.
Had the function been px^ (shownby the dottedcurve),he area
OQM would exactly represent the function.
To recapitulate : In the case of a function /("),if the curve
y =/(?"), (1)
and its firstderived curve y = f'(x)y (2)
be drawn, the rate of change of the function for any value of x
is represented equally well by the slope of the firstcurve and by
Y
Fi6. 18.
the ordinate of the derived curve for that value of a?; and the
function itself for any value of x is represented equally well by
the corresponding ordinate of the primary curve and by the area
of the derived curve increased by the constant quantity /(O).
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14-15] INTEGRATION INVERSE OF DIFFERENTIATION 33
The derived curve (2)is called the "curveof slopes" of the
first curve. Two of these curves are shown in Fig. 13. The
horizontal scale is the same for both curves ; but the ordinates
on the original curve represent lengths while the ordinates on
the derived represent tangents of angles. At a point at which
the original curve has a maximum or a minimum ordinate, the
slope is zero; and hence, the corresponding ordinate on the
derived curve is zero. Conversely, when the derived curve
crosses the avaxis, the corresponding ordinate of the original
curve is a maximum or a minimum.
ISi Integral curves. Let the curve whose equation is
y=f(^) (1)
be drawn. Suppose that the anti-derivative of f(x)is "l"{x)and
draw the curve whose equation is
y=fjf(x)dx,(2)
that is, y =
"f"(x)"^(0),
or, briefly, y = F(x), (3)
The curve whose equation is (2)or (3)is called the firstinter
grcU curve of the curve (1). It is evident that
^=^=/('). "
The following important properties can be deduced from equHr
tions (1),(2),(4).
(a)For the same abscissa x, the number that indicates the
length of the ordinate of the first integral curve is the same as
the number that indicates the area between the original curve,
the axes, and ordinate for this abscissa. Therefore, the ordinates
of the firstintegral curve can represent the areas of the original
Qurve bounded as above described.
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u INTEGRAL CALCULUS [Ch. II.
(6)For the same abscissa a?, the number that indicates the
slope of the first integral curve is the same as the number that
indicates the length of the ordinate of the original curve. There-ore
the ordinates of the original curve can represent the slopes
of the firstintegral curve.
Example. The line whose equation is
y=px
has for its firstintegral curve the parabola whose equation is
.y =
jpxdx,7?
that is, yz=p
At any point M on the ic-axis,OM being equal to x^^ say, erect
the ordinates MP, MPu to the line and the parabola. The same
/y.2
number, namely p-^,indicates both the length of the ordinate
MPi and the area OPM\ and the same number, namely pxi, indi-ates
both the length of the ordinate MP and the slope of the
tangent at Pi. This is true for the pair of ordinates erected at
every point on the a:-axis.
In like manner the curve whose equation is (2)has a first
integral curve. The latter is called the second integral curve
for the curve of equation (1). This second integral curve has a
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15-16.] INTEGRATION INVERSE OF DIFFERENTIATION 35
first integral curve which is called the third integral curve of
(1),and so on. There is thus a series of successive integral
curves for
any givencurve. For instance, the
second
integral
curve of the line y=px is the curve whose equation is
y
that is, y = P-^-
This curve is shown in the figure. Thesubjectof successive
integral curves has very important applications in problems in
mechanics and engineering. Accordingly, an exposition of their
properties and uses is given in Chapter XII.
16. Summary. This and the preceding chapter have been con-cerned
with showing by statement and examples that integration
may be regarded in two ways :
(1)As a process of summation, in which I f(x)dx denotes the
limit
ofthe sum indicated by
^ f{x)^x,whenAa:
approacheszero; "="
(2)As an operation which is the inverse of differentiation, in
which If(x)dx denotes d'^lf^x)x']r D-^f(x); that is,denotes
the anti-differentialoif(x)dx, or, what is the same thing, the anti-
derivative of f(x).
It may be remarked that the rules of integration are all derived
from the latter point of view. Both of these conceptions of
integration are employed in problems in geometry, mechanics, and
other subjects.The first view of integration is necessary to a
clear understanding of the application of the integral calculus to
the solution of certain problems; and, on the other hand, the
second view is necessary to a clear understanding of the use of
the calculus in the solution of certain other problems.
INTEGRAL CALC. " 4
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CHAPTER III
FUNDAMENTAL RULES AND METHODS OF
INTEGRATION
17. In Chapters I. and IT., the two purposes of integration
were set forth; and definitions of integration based upon these
purposes were given with illustrative examples. Relations be-ween
the definitions were also pointed out, particularly in Arts.
11, 13. It was also shown that in the process of making an
integration, whatever the
objectmay be, it is necessary to find
an anti-differential or an anti-derivative of some function. A
general method of differentiation is given in the differential
calculus. Unfortunately, no general method for the inverse pro-ess
of integration exists. It is necessary to derive a rule for the
integration of each function. The formulae of integration are
derived or disclosed by falling back upon our knowledge of the
rules of differentiation. In fact, the first simple rules, given in
Art. 18, are merely directions for retracing the steps taken in
differentiation. The inverse operation of finding an integral is,
in general, much more difficultthan the direct operation of find-ng
a differential or a derivative.
This chapter gives an exposition of the fundamental rules and
methods employed in integration. One or more of these rules
and methods will come into play in every case in which integra-ion
is required.
18. Fundamental integrals. Following is a list of fundamental
formulae of integration derived from the fundamental formulae
of differentiation. They can be verified by differentiation, as
36
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17-18.] RULES AND METHODS OF INTEGRATION 37
indicated in the first of the set. An additional list is given in
Art. 22.
Every integrable form* is reducible to one or more of the
integrals given in these two lists. The student should have
ready command of these formulae for two reasons : first,so that
he may be able to integrate these forms immediately; and second,
so that he may know the forms at which to aim in reducing com-plicated
functions. The functions to be integrated will not
usually present themselves in terms of these simple, immediately
integrable expressions ; and therefore a considerable part of this
book is taken up with algebraic and trigonometric transforma-ions
showing how to reduce given functions to these forms.
In the following formulae, u denotes any function of a single
independent variable.
J n + 1+ 1
in which n has any constant value, excepting " 1. The case in
which n = " 1 is given in II.
Differentiation of each member of I: with respect to u gives
w* du,
n.t j^=logt"co = logu + logc=:logct".
The different ways in which the arbitrary constant of inte-ration
can appear in this form, may be noted.
* ** An integrable form " here means a function whose integral can be ex-pressed
in a finite form which involves only algebraic, trigonometric, inverse
trigonometric, exponential, and logarithmic functions.
t According to I., lu-^ du =
-^+ c = - + c=oo + c. Nevertheless,
JJ " 1 + 1 0
_lu-^ du can be derived directly by means of I. For, on putting f-cin + 1
for c, which is allowable by Art. 9, \u^ du =^^^ + Ci. Now ^"
QJ
n+1 n+1
= - when n = " 1. Evaluation of this indeterminate form by the method of0
the differential calculus gives, differentiating numerator and denominator
as to n, M'+Uogtt, that is, logw. Hence iu-^du=\ogu +ci.
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38 INTEGRAL CALCULUS [Ch.III.
IT. Je~clw"~ + c.
y* rsin u dtt = " eoste 4 c"
VI. Jcos t*clt* = sin t* + c.
?!!" J8ec^i"clutanu + c.
Tin. rcsc*f*clt*-cotM
+ c.
IX. J8eciitanf"cliisecf" + c.
X. Jescucotiiclu=-csciic.
XI. ('--^^ = sin-iM + c =
-cos-iM + ci.
XII. r-^^ = toii-iw + c =
-cot-iti+ ^
/
"ci.
Xm. f ^= = sec-it* + c =-csc-iM + ci.
Jtt Vm2 - 1
XIT. f^"
= Ters-i t* + c = - coTers"* w + ci.
Ex. 1. fx"da;=.^^+c=}x*c.
J^ J"5
+ 1"4
4x*
Ex.8. fjC2^"JE.= f^a+s^"^)=log(i+ sinx)+c.
J 1 + sinx J 1 + sinx
Ex. 4. fx*(?x,
fxi'^cix,
fs""+"d",
{t^dt,(z^^dz,p^dp.
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18-19.] RULES AND METHODS OF INTEGRATION 39
EX. 5. j-. J'fIf,j-.- J-x-...-.-"^,g.
Ex. 6. fic*da;,
f"*d^ (x^dx,Jt?"dv, fa:"*(te,
fa;"*da;,
fx""(to.
Ex. 7. fVidaj,-^,f-^,fA^dx,f^cte, fVS?(to,r^.J J ^ J
y^J J J J J^
Jf' J"-l* Jie8+ 3' Juvw^-l' Ja"+ 2x2-2a; + 4'
tanx'
J sinx
Ex. 9. fc2"2(te, f*2*da;,(w + n)""to.
Ex. 10. fsin2xdf(2x),cos3x(f(3x), fseca4x(f(4x).
Ex. 11. fsecixtanixe^Cix),^^ . f~^^=, r_dOi5L
"^ "' VI-4x2 -^ VI - 9x-^ -^ Vl-tt2tttV
C 2dx C 3"to r 2dx r 4dx /* ^U/^"^^ ^^- J 1 + 4x"' J 1 + 9x2' J
2xV4^^^rT'.J4xVl6^^^3l' J
x^^^^^a2
19. Two universal formula of integration. In this article two
formulae of integration will be given which differ from those of
Art. 18 in that they do not apply to particular forms merely, but
are of a much more general character.
Suppose that /(a?),{x)y" (a?)," ", are any functions of x. Then,
A. j'{/(a5)l?'(a5)4"(a5)+-}"la5= (f(x)dx
-\-(jF(x)dx+(^(x)dx";
for differentiation of each member of A gives f(x)-\-F(x)
+"t"(x)\-
"-. This formula may be thus expressed : the integral
of the sum of any number offun"itionss equal to the sum of the
integralsof the several functions.
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40 INTEGRAL CALCULUS [Ch. III.
Ex.1. \{x^"m\x"e')dx=\x^dx"\mixdx"\ef^dx^
=
Jas'C08X
" e* +c.
Each of the separate integrations requires an arbitrary con-stant
; but, since all these constants are connected by the signs
+ and "
,their algebraic sum is equivalent to a single constant.
Again, if u is any function of a, and m is a constant,
B* \mu dx = fn\u dx^
for differentiation of each member of B gives mu. Hence, a
constant factor can be removed from one side of the sign of in-egrat
to the other without affecting the value of the integral.
It will soon be found that sometimes it will make the work
simpler to remove such a factor from the right to the left of the
sign I, and that, at other times, the process of integration will
be aided by putting such a factor under the integration sign. It
follows from B, that the value of an integral is unaltered if a
constant is used as a multiplier on one side of the sign J, and as
a divisor on the other. Thus,
udz
m
\udx = " \ mu dx =
mi
This principle will often be found useful.
Ex.1. J3x"i"3j-x"i"=ij-2x"i"f." + c.
Note. The value of an integral is changed if an expression that contains
X is transferred from one side of the sign i to the other. Thus,
(x^dx=:iofic;
but x\ xdx = ^si^-\-c.
Ex.2. f7.r*dfa;,az^Uz,(ad^ bxf^-^ dx, (a^^Ux, (^ ab^ cx^^-^ dx.
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10-20.] RULES AND METHODS OF INTEGRATION 41
Ex.8. f(x2_2x4-6)da;,(a^ 3"* + 6
x*)da.
Ex.4. (iZ+ btydt, f(a*-x*)"*K"f(costf+ sin tf)W.
Ex. 5. fe" dx, (e^dx, Tet*x, (e'^'dx,ef^+^'+^x2)dx.
Ex. 6. f(cosmx + einS x) dx, I{sec x4- coseo*^ (m + n)"}dx,
Ex.7, f^" '^,f ^^^
,f ^^
.
Ex 8 f "^^ r cto r dy_
JWT^^' J 9 + 4x2' J 16 + 264Vl-^
20. Integration aided by a change of the independent variable.
Integration can often be facilitated by a convenient change of
the independent variable. For instance, if f(x)dx is not imme-iately
integrable, it may be possible to change the independent
variable from x to t, the relation between x and t being, say
X = ^{t)yO that f(x)dx is thereby put into a form F(t)dt which
can be easily integrated. Experience and practice afford the only
means of determining the substitutions that will be helpful in
particular cases. The actual substitution of the new variable
may often be conveniently omitted, as in Exs. 1, 2, 3, below.
Ex. 1. ('(xa)"dx.
On putting x-{- a=^t^ dx = dt\ and the given integral becomes
J n+1 n+1
Since dx = d(x + a),the given integral may also be written
f(x+
aYd(x
+
a),and x + a being regarded as the variable ti, the integral
is (^+ ^)"'''4-c,s before.
n+1
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42 INTEGRAL CALCULUS [Ch.Ul.
Ex. 2. r8ec2(6-2a;)da;.
If 5 - 2a; = ", da; = -
Jd" ; and
fsec* (6 - 2x) da; = - l^B/^Hdt - Jtan " + c
= - Jtaii(6-2x)+c.
Since da; =
--Jd(6" 2
a;),heintegral may be written also
fsec2 (6 - 2x)da;= - J fseca(6 - 2a;)d(6- 2x)
=
-itan(6-2x)+c.
Ex. 8. fc"+""da;.
If a 4- fta; f, da; =
-dt^ and"
Ce"+"'da;-fcdf le"
+ c =i6"+""
+ c.
Otherwise : since da; =i d (a + "a;),"
f""+""da; =
i fe"+*'d (a + 6a;)
ie"+"" + c.
TT^ " C 12x2-4a;
+ 6^"^^"*-
J4xs-2x24-6x-10^-
On putting 4x8 - 2x2 + 6x - 10 = ", it follows that (12x2-
4x + 5)dx= dt^ and
Note, /if "/i"expression under the sign of integration is a fractionwhose
numerator is the differentialofth^ denominator^ the integral is the logarithm
of the denominator.
Ex. 5. f^-"^.J sin"*x
If sin X = {, cos xdx = d^ ; and
J sin^x J (^ bt^ Ssin^x
Otherwise :f"2!.*i5
^("'"")
=1_
+c.
J sin^x J (sinx)^ Ssin^x
The necessity of learning to recognize /orws readily will be apparent
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20.] BULBS AND METHODS OF INTEGRATION 43
Ex. 6. f-^^-.
If x +
then ("^%= ^^^^^= f(--4 l-iW
If X-\-I
=: Zj dx = dz^ X = Z " If
Ex.7. f ?^(te.
If e* + 1 = 0, e*da; = d^;, e* = Z'-l;
andf_^_c?x=f-^:i!^= f(^lli)tf^=f(,i_;,-J)^
"^(e"4-l)* ''(6-+ l)i -^
."*
-^
= A("*+l)*(3e"-4).
f(2+ 3a;)*da;,f(3-7a;)*dx.
Ex. 9. \cos(x"\-a)dx,\aec\x+a)dx,\ "
^.^
^ .y\sm(a-\-hx)dx.
J J'
Jcos'*(4 O 05)
J
Ex.10. fcos^cZx,e2+""da;,i'^dx,f^^dx.
Ex. 11. fx(a + ")*d^,
f ^^^" *
"*
^*^ (a + 6x)*
Ex 12 f ^ rcos(loga;)da; r d^*
J (l + fl;2)an-la;'J a;
'
"^sin^f^'j.
Ex.18. {{a-^hz)'^dz,yJia^hxydx, {-r7=====:
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44 INTEGRAL CALCULUS [Ch. III.
Ex.14, f,
^" (Put2x-l=4jp.)
J Vl6 + 4a;-4a;"
Ex. 15. f22!9"^. (Put sin X = ".) f(sm*^ - 8 mn*e + 4 abcfleJ siii"x
^
+ 11 sin tf+ 2)cos$d$, f(tan"0 - 7 tan"0 + 2 tan^ + 9)sec"0d^.
2L Integration by parts. Two universal formulsB of integra-ion
were given in Art. 19. A third formula of this kind will
now be discussed. Differentiation
showsthat, u
and
v being any
functions of x,
d/ N
du.
dv
This may be written also in the differential form,
or more simply, d (uv) vdu + udv, (1)
in which, du
=-^dx,s^nd dv =
-^dx.x dx
Equation (1)becomes on transposition,
udv =
d(uv) vdu.
Integration of both members of this equation gives
"" \udv = uv " \v du.
Equation C may be used as a formula for integrating udv when
the integral of v du can be found. This method of integration,
commonly called" integration by parts," may be adopted when
f(x)dx is not immediately integrable, but can be resolved into
two factors, say u and dv, such that the integrals pf dv and v du
are easily obtained. The procedure is as follows :
if{x)dx= iudv]
whence by C, = wv " I v du.
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20-21.] RULES AND METHODS OF INTEGRATION 45
No general rule can be given for choosing the factors u and dv.
Facility in using formula C can be obtained only by practice.
This formula has a greater importance and a wider application,
than any other in the integral calculus. The following examples
will show how it may be employed.
llogxdx = X log X " t X"
= xlogx " X4- c.
Ex. 8. Find fx"*dx.
Let tt = e", '^^'=
xdx ;
then, du = e'dx, " = J x^.
The formula gives
Cxe'dx JxV - J fxVdte.
But x^"i*dx is not so simple for integration as xe*dx. This indicates that
a different choice of factors should have been made.
On putting" m = x, dv = "*dx,
du = dx, " = e*,
and formula 0 now gives
\X"i'dx = xe'" le'dx
= xe* " e* + c
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46 INTEGRAL CALCULUS [Ch. III.
Ex. 4. Find (ufie'dx.
Let w = a^, dv = e'dx ; then du = Sx^dx, r = c*.
Hence, (ofie'x = ofie'- sCx^e^x.
To find fxVda;,put u = a;^, ^v = e'dx; then du = 2x"te, " = e* ; and
(x^e^dxaV - 2 Cxe'dx,
By Ex. 3, (xcdx= X6* - c*
-fc
Hence, combining the results,
(x^e'dxe*(a!"
3x2 + 6x - 6) + c.
This is an example in which several successive operations of the same
kind are required in order to effect the integration. Many such examples
will be met, and usually a formula called " a formula of reduction "will be
found for integrating them. ** Integration by parts" is of great use in
deducing these formulae of reduction. In order to avoid making mistakes in
cases like Ex. 4, a good plan is to write down the successive steps in the
integration clearly, without putting in the intermediate work, which can be
kept in another place. Thus :
fx8e" "Zx = x8e" - 3 fxV dx
= x8e* - 3 [xV - 2 fx"* dx2
= x8c" - 3 [x2e" 2 (X6* e*)]4- c
=
e*(x8-3x2 4-6x-6)4-c.
Ex. 6. jsin~i X dx.
Ex. 6. I cot-i X dx.
Ex. 7. (za'z.
Ex. 8. (x^a'dx.
Ex. 9. Itan"i x dx.i
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21-22.] BULES AND METHODS OF INTEGRATION 47
Ex. 16. fsec^ X log tAn xdx. Ex. 17. fsee"** dx,
Ex.16. fa;"(loga:)2(to. Ex.18. fx""logx(te.
22. Additional standard forms. Some fundamental integrals
which often appear are collected together in the following list.
Their derivation will be found in the next article.
XY. rtantt"lt"log8eeu+ C.
XVI. fcot udu = lo? sin w + C.
XVII. j'sectt"lwlogrtanf||Wc.
XVIII. Jcosee"It" = logrtan ^+ C
XX. f-/!i_=ltan-i^c =
-lcot-i^+C'.
XXIL f^=^^=:r =
yer8-i^+C-coyer8-i^+C".
XXIII. f_^*?L-=
-Llog ?^^=^+ C =
i-tonh-i+ C.Ji^2_"i2 2at"H-a a a
XXIV. f-"^^=: = log(tt+ "/fi2Ta2)+C =
sinh-i^+C'
XXV. f ^^= log (w + "/tt" a") + C = cosh-i
^+ C.
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48 INTEGRAL CALCULUS [Ch.lU.
23. Derivation of the additional standard fonns.
Formula XV. Since tan u = ^iBif,cosu
J J cosw"/
cost*
= " log COS u = log sec u,
Fonnula XVI. fcotdu= C^^du =C^Sm^
J J suiu "/ sinu
= log sin 1* = " log cosec u.
Formula XVIII. Since cosec u = cosec u^Qsecu-cotu
cosec tt " cot It
/"."""/", ^-.^ r" cosec tt cot tt + cosec* u ,
,
cosec uati= I ' duJ cosec tt "
cottt
/d(cosec " cot u)cosec M " cot u
= log (cosec " cot u)= log
cosec u " cot tt
1"cost*
sinu
"
28in"|
2sm^co8^^
Formula XVII. On substituting w +|for w in XVIII., there
results
J*cosecu
+^du
log
tan^|4-|\'
that is, Isec udu = log tan ( 4-j )
Formula XIX. If u = aay then du = adz, and
"
"1
*"
-1
*"
= Sin*
- = " cos*
"
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23.] RULES AND METHODS OF INTEGRATION 49
Formula XX. If w = az, then du = adz, and
1,,tt
1.
,tt= - tan~^ - = cot"^ "
a a a a
Formulae XXI., XXII. These integrals also can be derived by
means of the substitution u=
az.
Formula XXIII. Since"1"
=
-^(" -^\" "r 2a\u " a u + aj
J u* " 0? 2 aJ \u " a u + aj
= A Uog (w -
a)- log (w+ a)}
2a u + a
Formula XXIV. If u^-\-a^ = s?, then udu " zdz,
du__
dz
z u
Hence,
and, by composition.u-\-z
Therefore, f^^
=f^?^
"/ Vtt^+ a^J t* + z
= log (tt 2;) c = log (u+ V w* + a*)+ c
= log!i"^^"^ + c. = 8inh-"'i+ c.
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50 INTEGHAL CALCULUS [Ch. IIL
The latter result can be derived also in the following way :
On putting u = az,
r_du_^ r-^= sinh-^ + c
= sinh-* - + c
a
Formula XXV. Similarly, on
puttingi** " a* = a*,
= log h c, = cosh"* - + (^
Or,putting
" = cw;,
f /^ = r-^==cosh-^4-c = cosh-*!*+ c
Ex. 1. Find fVaa-x^da;.
Integrating by parts, let
tt = Va2 " "'", dv = dx.
Then dti =
^
(Zi;, v = x,
andfVgg - a;g(fa= xVg^ - x^ + f
"
^Jg~.
.
Since V^aZT^ =
^^ "" ^^
,it follows that " ^^ =
^^- Va^- x^.
Vo2Z^ Va2-x2 Va*^-x2
Hence, fVa^ - x^dx = xVa" - x^ 4- a^ f ^ f Va^ - x^dx.
From this, on transposing the last integral to the firstmember,
fVa-2-x2(toi /xVa2 - x^ + a^sin-i ^V
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23.] RULES AND METHODS OF INTEGRATION 61
Ex. 3. r ^ /^ f ^
=
if^
=sin-i f^-^-^V
jj^ 3r dx
_
r dx
"'Vxa4-8x4- 62JV(x 4- 4)"
-f36
= log (X 4- 4 4-Vx2 4-8x4- 62).
Ex. 4. f ^^=f
^ =
-JLtan-i^+^.x2 4-6"4-12 J (X 4- 3)24- 3 V3 \/3
3)~2Ex. 5. f ^
=f^
=llogI^"
Jx*4-6x4-6 J(a;4-3)"-4 4'^
(x 4- 3)4-2
=hog^"i.4 *x4-6
Ex. 6. f ^" Ex, 15. f-^-.
"^ \/x2-6x^ a 4- x"
\/x2-6x
y/fdxx. 7. f ^^^.
Ex.16, f-^.^ V7x*4-19
J y" - 8
Ex. 8. f
^^^^Ex.17,
f-^^."^ V3x*4-2x2-l
^ tan ox
Ex. 9. f "
Ex, 18. fcot (ax 4- b)dx.
"^ v4x " x^
Ex 11 f ^ Ex, 80. f^
r= f ^(^-2) ].JV43^* - x-^-4x4-8L
J(x-2)H4j
Ft 12 f_^?_.
Ex, 21. f^
^
^'^^^'
JVT^Jxa-4x-8
Ex. 18. f6tan3x(te. Ex. 88. J(sec2 x 4- 1)""te.
Ex, 14. f-4^=- Ex.83, f ^
"^\/3-6x3 "'(x--a)V(x-a)2~6"INTEGRAL CALC. " 6
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52 INTEGRAL CALCULUS [Ch. III.
Ex. 24. f ^Ex. 89. f ^
Ex. 26. f-^^. Ex. 80. (l"^^de,"^ V x* " c* J sin ^
Ex. 86. ^xyV^fi^dx.^^ ^^
r dx
^yJaH^ + 2 6a;+ c
Ex.87, f ^Jl
Ex.88, f ^
^ Vl6 X " ttx'* [Rationalizehe numerator.]
24. Integration of a total differential. It has been shown in the
differential calculus, that if
"=/(".y). (1)
Xy y, being independent variables, the total differential of u is
equal to the sum of its partial differentials with respect to a; and
y. That is,
(ltt ^d"+ ^dy. (2)ox ay
It will be remembered that when differentiation is performed
with respect to a?, y is regarded as constant, and when differenti-tion
is performed with respect to y, x is regarded as constant.
Suppose that a differential with respect to two independent
variables is given, namely,
Pcbi+Qdy, (3)
in which P and Q are functions of x and y. The anti-differential
of (3)is required. Not every function (3)that may be written
at random has an anti-differential. Hence, it is necessary to de-ermine
whether an anti-differential of (3)exists or not, before
trying to find it. It has been shown in the differential calculus
that if u and its first and second partial derivatives with regard
to Xy y are continuous functions of x, y, then.
dy dx dx dy(4)
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23-24.] RULES AND METHODS OF INTEGRATION 53
If (3)has an integral, say w, then,
du "
Pdx-\-Qdyj (5)
in which P=$^, (6)ox
and Q = g. (7)
Differentiationof
both
members of (6)and (7)with respectto
y and x, respectively, gives
dP d^u
dy dy dx
dx dx dy
Hence, by (4), |=^. (8)
Therefore, if (3)has an integral, relation (8)holds between the
coefficientsP, Q,and the differential (3)is then said to be an exact
differentialConversely, it can be shown that if relation (8)
holds, the differential (3)has an integral. For the present the
latter proposition may be assumed to be true.* The condition
(8) is called the criterion of the integrability of the differen-ial
(3).
Suppose that the coefficients P, Q satisfy the test (8),then
there is a function u which satisfiesequation (5). Since Pdx can
have been derived only from the terms that contain a?,integration
of the second member of (5)with respect to x gives
fPdx +c.
in which c denotes any expression not involving x.
"For proof, see Introductory Course in Differentialquations, Art. 12,
by D. A. Murray (Longmans, Green " Co.).
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54 INTEGRAL CALCULUS [Ch.111.
Now Q dy has been derived from all the terms of u that contain
y. Some of these terms may contain x also ; and if so, they have
been discovered already in jPdx, Therefore, the remaining
terms of u that do not contain x will be found by integrating
with respect to y the terms of Qdy that do not contain x. Hence,
the following rule: Integrate P da? as if y were constant; inte-rate,
as if a? were constant, the terms in Qdy that do not contain
X
; add the results andthe
arbitrary constant ofintegration.
Ex. 1. Integrate ydx-{-xdy,
dP dOHere P = y, Q = x; hence ^ = 1*
-^= 1" and thus, criterion (8) is
satisfied.
Also, iPdx = iydx = xy; and there are not any terms in Q dy without x.
Hence the integral isxy + c.
Ex. 2. Integrate ydx " xdy,
riPdO
Here P=y, Q= "x;hence " = 1,
-^= " 1, and the criterion is not
satisfied. Therefore an integral of the given expression does not exist.
Ex. 8. Integrate (x^ ^xy - 2^) dx-\-(^
- ^xy - 2x2) dy.
Ex. 4. Integrate (a^-2xy-y^)
dx- (x + y)^dy,
Ex. 5. Integrate (2 ax-\-
by"\-g)
dx"\-(2ay + bx + e) dy,
25. Summary. The directions so far given for obtaining the
indefinite integral of f(x)dx may be summarized as follows :
(1) Memorize the fundamental formulae of integration, given
in Arts. 18 and 22.
(2) Acquire familiarity with the application of the principle of
substitution, or change of the independent variable, discussed in
Art. 20.
(3) Use the firstand second universal formulae of integration,
A, By given in Art. 19.
(4) Learn to apply with ease the third universal formula of
integration, namely, the formula for integration by parts given in
Art. 21.
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56 INTEGRAL CALCULUS [Ch. III.
10. f "f^
+.
^ Va;,n(sm2^ + cos30)d^,
J""toLVa;2 + a2 - Vi^^^aJ
11- f:===" (Pat a; = -. Compare result with formula XXI)
f
_^.
(Put sin a; = "f. Compare with XVII.)
12. f ^(Put3+2a;=a?.)f
-^-. (Putl+a;="2.)
JV1-3X-X2
"^(i+a;)f+(i+^)J
18. r^i^l^dir.^^ . (Puta + 6x2 =
;52.)*^ 2 "" "^ Va + 6x2
14. Jsec-ixclx,cosec~ix(te,cos-ix"lx,x2sin""ix(te.
16. Ixcosxdx, Jx^sinxdx,xtan2xdx.
16. fx'a'dx, Tsui log cos X(ix, rcosec2xlogcotx(lx.
17. f(2x-2)(x2-2x+ 6)cos(x2-2x + 5)fZx. (Putx2 -2x + 6=;?.)
18. fx61og(x"+ a8)dx. (Putx8 + a8 = ".)
19. I ^ ^^ dx, (Integratey parts, putting u = (logx)2.)"^
x^
20. Show that
f(logx)"dx=
x[(logx)"-
m(log x)"-i+ m(m" l)(logx)""-* "
+ (- 1)""im(m- i)...3.21ogx+ (-l)".ml].
21. Show that
fe"x" dte= x"e" - m fe*x"-i dx =
6*[x* mx"" -1 + m(m" l)x""-" """
+ (-_l)"-%(m-l)...3.2.x + (-l)"".wl].
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25.] RULES AND METHODS OF INTEGRATION
22. r(sec + cosec xy dx.
28. f-i*-J 1 + COot^
dx24
-x^
r dx
J Va2 + 62 _-
25. Evaluate 2 f^log
tan 0 . cosec 2ede + log cot ^f cosec 2 ^ "W.
26. f(a
tan ^sec ^+ 6) cot ^"W.
"^V2 + 2a;-"2 J V- 16 -
28. flog("+Vx2-a2)^
dx_
lOx-i
(2x
29. flog(" + VS^Tl?) "^.
30. jJ'^^^de.
g-rasecgg + ftcosec^g^p
3gfsecxtanxda;
* J tan^ + cot^
* '
J tan2x-2
oo C 1^ r. 39. f" ^.2. jvers-i -'Vxdx. J e" " 6 e"*
^ ^ 40 f "^^
88. jV"Ta^dx,'
Jy/a^-b
r ^^M^ r sinxdx
"^xi-5xiJsm-Vcosx
85. f^-^ 42. f '-
Jsinx+.cosx
.Jax2 +
2
dx
6x + c
36. f,
^^.
" 48. f ^
37.r(3x + 4)V^+T^^^ 44.
r
c
dx
V2 " X. V- ax2 + 6x + c
45. Integrate sin x cos y dx + cos x sin y dy.
46. Integrate cos x cos y dx " sin x sin y dy.
47. Integrate (3x2 + 6xy + 4y2)dx +(3x2 + Sx^^ + 6y2)dy.
67
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CHAPTER IV
GEOMETRICAL APPLICATIONS OF THE CALCULUS
26. Applications of the calculus. In this chapter some of the
practical applications of the integral calculus are discussed. In
particular, the areas of curves and the volumes of solids of
revolution are determined. Art. 32 deals with the deduction
of the equation of curves from data whose expression requires
the use of differential coefficients.
There is one common
aimin by far the larger
number of the
simpler applications of the integral calculus. This aim is to
find the sum of an infinitenumber of infinitely small quantities.
The process of summation has been discussed in Chapter I.
The student will find that in most of the problems there are
two steps to be taken in order to obtain the solutions, viz.:
(a)To find the
expression
for
any
one
ofthe infinitesimal
quantities concerned and to reduce it to a form that involves
only a single variable;
(b)To integrate this differential expression between certain
limits which are assigned or are determinable.
Each of the differential expressions is called an element, "
an element of area, an element of length, an element of volume,
an element of force, etc., as the case may be.
27. Areas of curves, rectangular coordinates. It has been
shown in Arts. 3-5 that the area* between the curve y=f(x),
* The calculation of such an area iscalled** Quadrature of curves." From
this comes the phrase ^^to perform the quadrature," which is often used as
synonymous with **to integrate." The areas of only a few curves could be
found before the discovery of the calculus. Giles Persone de Eoberval (1602-
68
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26-27.] GEOMETRICAL APPLICATIONS 59
the o^axis, and two ordinates for which x = ay xssb, is ex-pressed
by
f/(x)dx.It has also been shown that this area can be evaluated by
finding the indefinite integral
of f{x)dx,substituting b and a
in turn for x in the indefinite
integral, and taking the differ-ncebetween the results of the
two substitutions.
Ex. 1. Find the area bounded
by the parabola whose equation is
y^ = 4 ox, the axis of x, and the
oi*dinate at a; = xi. Also find the
area between the parabola y^ = 9x^
the axis of x, and the ordinates for
which " = 4, jc = 9.
Let QOC he the parabola whose equation is y* = 4ax. Take 0M= xi,
0A = 4, CD = 9; erect the ordinates MP, AB, DC, Suppose that two
ordinates B8, VT are drawn at a distance dx apart. The element of area,
which is the area of any infinitesimal rectangle like BSTV, \b ydx. The
area required in the first case is equal to the sum of the areas of all such
rectangles, infinite in number, that are between OY and PM; that is, be-ween
the limits zero and xi for x. Hence,
Fig. 16.
area of 0PM = [\dx.First of all, y must be expressed in terms of x. This can be done by
means of the equation of the curve, from which
y = " 2a^xi
1676), professor of mathematics at the College of France in Paris, Blaise
Pascal (1623-1662),ohn WalUs (1616-1703),avilian professor of geometry
at Oxford, considered an area to be made up of infinitely small rectangles,
and applied the principle to the determination of the areas of parabolic
curves. The French geometers iound the formula for the area between the
curve
y= x", the
axis ofx,
and any ordinatex = h
whenm is a
positiveinteger. Wallis found the area when m is negative or fractional. This was
before the development of the calculus by Leibniz and Newton.
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60 INTEGRAL CALCULUS [Ch. IV.
Hence,
that is,
(The positive sign denotes an ordinate above the x-axis, the negative, one
below.)
area of 0PM =
("^2aMdx
area of 0PM = two thirds of the area of the circumscrib-ng
rectangle OLPM,
The area OPQ = 2 0PM = two thirds the area of the rectangle LPQR,
In the second case :
area-4J5C2)=f
ydx
= 3 ("Veto3 [f"* + c]9= 38.
If the unit of length is an inch, the area of ABCD is 38 square inches.
Ex. 2. Find the area between the curve 2/^= 4 ax, the axis of y, and the
line whose equation is 2/= 6.
X
_A_
B
Fio. 16.
In this case it is more convenient to take for the element of area the
infinitesimal rectangle indicated in the figure. The element of area is thus
xdy\ and
area OAB=i \ xdyJo
Jo^a
12 a
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27,] GEOMETRICAL APPLICATIONS 61
Ex. 8. Find the area of the ellipse -5 + 7^= !"
The area required isfour times the
area of the quadrant A OB. An ele-ent
of area is the area of an infini-esimal
rectangle BSTV, namely
y dx. The sum of allthese elements
from O to ^ isexpressed by \ ydx,Jo
From the given equation,
Fig. 17.
a
in which the positive sign denotes
an ordinate above the x-axis, and
the negative sign, an ordinate be-ow.
Hence,
area of ellipse= 4 OAB = 4 \ydxJo
aJQ
which by Ex. 1, Art. 23, = ^ [|S^^^^a + fsin-ic]*= Tab,
If a = 6, the ellipse is a circle whose area is ira^.
Find the area included between the ordinates for which " = 1, a; = 4, the
curve, and the axis of x.
Area PQMN= ^^ydx ^a^ - x^ dx
="rEVS2T:^
+ ^sin-i^cT12 2 a Ji
= 5|2Va2_l6iVoaTTT + ^fsin-i^sin-ii^If the semi-axes are 5 and 3,
area PQMN = f{6 - V6 + " (si^"^ - si^"^ *)"
= f{6- 2.454)+ ^ (.927.201)
= 3.778.
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62 INTEGRAL CALCULUS [Ch. IV.
Fxo. 1"
Ex. 4. Find the area between the
curve whose equation is
y =
A(aJ-l)(aJ-3)(x-6),
the axis of x, and the ordinates for
which X = " 2, a; = 7.
7
The area required z^\ydx
"27
-
^C(pfi
9a;2 + 23"-
16)"to
Further remark on this example
may be instructive. On putting
y=
0, the intersections of the curve
and the x-axis are seen to be at the
points for which x = 1, 3, 5. That
is, referring to the figure, OC = 1,
0Z" = 3, 0E = 6.
1-i
Area ^P(7= fydx = - W-
This area appears with a negative sign, since the ordinates are negative in
APC because itis below the x-axis.
Area CHD = Ty dx = + J,
the sign coming out positive since CHD is above the x-axi"
ATea,DLE=(ydx =
-\;
and.
area EQB = \ ydx = + S.
The area required = area APC + area CHD + area DLE + area EQB
The absolute area=W
+ i + i + 3 = 12^.
as obtained before.
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27-28.] GEOMETRICAL APPLICATIONS 63
This is an example of the principle indicated in Art. 5, namely, that when
the area between a curve, the x-axis, and any two ordinates, is found by inte-ration,
this area is really the sum of component areas, those above the
X-axis being affected with the positive sign, and those below the x-axis with
the negative sign. The next example will also serve to illustrate this.
Ex. 5. Find the area between a semi-undulation of the curve y = sin x
and the x-axis.
The curve crosses the x-axis at x = 0, x = ir, x = 2 ir, etc.
Area of ABC = \y dx = f'^sinx dte = [- cos x + c]'= 2.
Jo Jo "
But BTea,ABCDE=C''ydx=C''6mxdx 0.Jo Jo
The total area, regardless of sign, is 4.
Y
Fie. 19.
28l Precautions to be taken in finding areas by integration. The
method offinding
areas whichhas been described in
the last
article can be used immediately and with full confidence in the
case of a curve y=zf(x),only when the limits a and b are finite,
and the function f(x)is continuous and one-valued for values of
X between a and b, and does not become infinite for any value
of X between a and b. Special care must be taken in cases in
which anyone
ofthe
conditions justmentioneddoes
nothold.
While, in some of these cases, the application of the method of
Art. 27 will give true results, in other cases it will give results
that are altogether erroneous. A few examples are given below
in order to emphasize the necessity of caution.
Ex. 1. There is a double value for y in the parabola y^ = 4 ax. This was
considered in Ex. 1, Art. 27.
Ex. 2. Find the area included between the parabola (y " x " 6)2= x,
the axes of coordinates, and the line x = 6.
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64 INTEGRAL CALCULUS [Ch. IV.
In this case y = " " Vx + o ; and thus to each value of x belongs two
values of y. The ambiguity can be removed by defining more exactly what
area is meant. If the area OBPM is desired, the value of y corresponding
to each value of x between 0 and 5 is x " Vx + 6. Hence,
area OBPM= (\x - Vx + 6)dx
V M
Fio. 20.
If the area OBQM is desired, the value
of y corresponding to each value of x be-ween
0 and 6 is x + Vx + 5 ; and hence,
areaOjBQJIf=f% + Vx + 6)dx"/o
=
V- + ""/^.
If the area PBQ^ between the curve and the line x = 6, had been required,
it would have been necessary firstto determine the areas OBPM, OBQM.
Area PBQ = area OBQM- area OBPM;
= ^V5.
Another way of finding the area of PBQ is the following. Let TS be any
infinitesimal strip of width dx parallel to the y-axis. Evidently, TS is the
difference of the values of y that correspond to x= OV. Hence, denoting
these values of y by yi, ^2,
area PBQ =\(yi- y^)dx
Jo
= 1{(x+ Vx + 6)-(x-Vx + 6)}dx
= r 2Vxdx
= ^V6.
Ex. 8. Find the area included between the witch y =
o"
and its
x2 + a2
asymptote. The asymptote is the axis of x, and hence, the limits of integra-ion
are
+oo
and" oo. In this case it is
allowableto use infinite limits.
Eor, on finding the area OPQM between the curve, the axes, and an ordi-ate
at distance x from the origin,
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28.] GEOMETRICAL APPLICATIONS 66
area OPQM= CydxJo
Jo "2 + a2
rrfanan-i^H-cTa Jo
= a2tan-i^.a
Fig. 21.
If the ordinate MQ be made to move away from the origin towards the
right, that is, if the upper limit x increases continuously, then tan-i-
increases continuously, and approaches - as a limit. Hence,
2
r^0
o
x"^ + a^
ira-*
represents the true value of the area to the right of the y-axis. Since the
curve is symmetrical with respect to the ^-axis, the area required is double
this, namely, va^,
Ex. 4. Find the area included between the curve y* (x^"
a^y= 8sc*,the
a;-axis, and the asymptote x = a.
2x
In this case y = "
To every value of x corresponds a real value
of y ; but, when " = a, y is infinite. Therefore a special examination is re-quired.
For values of x less than a, however, y is finite. Then, for a: " a,
area OMP2xdx
= (
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66 INTEGRAL CALCULUS [Ch. IV.
As X approaches the value a, it is apparent that the area OMP approaches
3 a' as a limit ; and hence,
.
Fio. 93.
Ex. 5. Find the area bounded by the curve in Ex. 4, the x-axis, and the
ordinate at x = 3 a,
It has already been noticed in Ex. 4, that /(") becomes infinite when
x = a. Asa lies between the limits of integration, 0 and 3 a, the integration
formula for the area should not be used until its applicability is determined
by a special investigation. The area from x = 0 to the infinite ordinate at
X = a, has been shown in Ex. 4 to be 3 a'. The area to the right of the
ordinate at x = a will now be discussed. Since /(x) is finite for values of x
greater than a, then for limits, x " a and x = 3 a,
area Jf'P'CiVirf-2xdx
^i
= 6a*-3(x2-o2)*.
As X diminishes and approaches a, this area approaches 6 a* ; and hence,
the area between the infinite ordinate at x = a, and the ordinate at x = 3 a,
is 6ai.
Hence, the total area between the curve, the x-axis, and the ordi-ate
atX = 3a, is 3 a"
+6 a", that is Da'.
The same result is obtained when I " is evaluated in the ordi-
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28-29.] GEOMETRICAL APPLICATIONS 67
nary way ; and thus, the integration formula for the area holds good in this
case, although /(x) becomes infinite for a value of x between the limits of
integration.
Ex. 6. Find the area included between the curve y(x"
a)^= 1, the axes,
and the ordinate x = 2a.
Immediate application of the integration formula gives for the area,
r^ dx_
r 1_
_,.gl2"_
_
2
Jo (X -
a)2L X - a Jo a
FiQ. 28.
But, /(x), which is the length of the ordinate y, becomes infinite for
x = a; and, if an investigation be made similar to that carried out in Exs.
4, 5, it will be found that the area is infinite. For, OM being equal to x,
area OMPB= C ^, =
-iL
Jo (x "
a)''^a" x a
It is evident, that as x increases from 0 to a, the area increases from 0
to CO. Consequently, the area between the curve, the axes, and the ordinate
at X = a is infinite. Similarly, it may be shown that the area between the
curve, the x-axis, and the ordinates at x = a, x = 2 a, is infinite. Therefore
the total area required is infinite. Hence, the integration formula for the/"2a
f[xarea, namely, i ? fails in this particular case in which f(x) be-vo (x-a)2
^*"
^ ^
comes infinitefor a value of x between the limits of integration. This con-clusion
may be compared with that in Ex. 5.
29. Precautions to be taken in evaluating definite integrals. It
has been shown in Art. 6, that any definite integral, say jf{x)dXj
maybe
graphically representedby the area between the curve
y =/(aj),he axis of aj, and the ordinates at a? = a, a? = 6. Hence,
INTEGRAL CALC. " 6
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29-30.] GEOMETRICAL APPLICATIONS 69
30. Volumes of solids of revolution. Let PQ be an arc of a
curve whose equation is y =f(x).Draw the ordinates AP, BQ,
and let OA = a, OB=b, The vohime of the solid PQML
generated by the revolution of APQB about the a;-axis is re-quired.
On the revolution of FQ each point in the-arc PQ
will describe a circle. Suppose that AB is divided into n equal
parts Ax, and let OQi=a?, QiQ2 = Ax, Construct the rectangles
^1^2? P2Q1 as indicated in the
figure, and suppose that they
have revolved about OX with
APQB,
It is evident that the volume
of each plate, such as PiPj-^g-^u
of the solid of revolution is less
than the volume of thecorre-
sponding
exterior cylinder gen-
.eratedby the revolution of the
rectangle Q1R1P2Q2 about the
aj-axis,and greater than the vol-me
of the corresponding interior cylinder generated by the
revolution of
the
rectangleQ1P1R2Q2 about the aj-axis. Now,
the volume of the cylinder generated by Q1P1R2Q2
Fig. 24.
=
ttFiQiAaj
and the volume of the cylinder generated by Q1R1P2Q2
=
TrP2Q2Ax
=
7r[/(ajAaj)PAaj.
Hence, w [f(x)YAx" PiP2^2^i " ^ [/(x+ Ax)yAx,
Suppose that PQML is divided into n plates, such as P1P2N2N1,
one plate corresponding to each segment Aa; of ^B; and suppose
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70 INTEGRAL CALCULUS [Ch. IV.
also that the interior and exterior cylinders corresponding to
each of these plates are constructed. Then, on taking the sum
of all the interior cylinders, and the sum of all the exterior
cylinders, and the sum of all the plates P^P2N2Ni, the latter
sum being the volume required,
VTT [/(a?)]2Aa;PQML " V ^ [/(^+ Aaj)]2Aa?.
X = a 36 = 0
As Aa? approaches zero, the sum of the exterior cylinders ap-roache
equality to the sum of the interior cylinders. The
difference between these sums is at the most an infinitesimal of
the first order when Aa? is an infinitesimal, and accordingly has
zero as its limit. Therefore, since the volume required always
lies between these sums,
a;=6
volume PQJfL = limit^
V"^[/(")?Aa;;
that is, volume PQML = (\ [/(a?)] dx.
The element of volume is tt [/(a;)]x ; this is usually written
Try^dx, since y =f(x).This value of the element may readily be
deduced from the figure on supposing that Q1Q2 is an infinitesimal
distance.
If an arc of y =f{x)between the points for which y = c and
y = d revolves about the y-Sixis, it
can be shown in a similar way that
the element of volume is wo^dy, and
that the total volume generated by
the revolution is
TTj^^^y-
Before integrating it will be neces-sary
to express x^ in terms of y.
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30.] GEOMETRICAL APPLICATIONS 71
E: '1. Find the volume of the right cone generated by revolving about
the X: 'is the line joiningthe origin and
the poiii (Ji^).Let M be the point (A,a).
The equation
of OiJf is
ax = hy.
The elemt it of volume iswy^
dx. Hence,
velum i OMN = TT I 3
Jo h^
y'^dxI
3Fio. 26.
This may be interpreted : the volume of the right circular cone QMN is
equal to one thi 'ithe area of the base by its altitude.
Ex. 2. Find 'he volume of the cone generated by the same line on revolv-ng
about the y-^xis.In this case, he ejement of volume
is TTX^ dy. Heni 3,
volume OM V= ir 4 x'
^0
Jo a'2
3
Ex. 8. Find the ^ olume of the solid
generated by revolv.ng
the arc of the Fig. 27.
parabola y^ = 4px t :tween the origin and the point for which x = xu about
the X-axis.
In this case,
volume OPPi = ir ('\^dxJo
/"Xl
= IT I 4pxdx
Jo
= 2irpxi^;
or, since yi^ = 4pxi,
Hence, the volume is one half the
volume of the circumscribing cylinder.
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72 IXTEGUAL CALCULUS [CH.r*;V.
Ex. 4. Find the volume generated by revolving the arc in Ex. 3 alx * (^xxtthe
y-axis.^
In this ca^e,,(
volume OPQ =
ttCx^i ^ly
f" "
, y*
or, since yi^=
4pz'-.
volume OPQ = ^ t \jfiXi^,of the 't oy
ence, the volume required is one fifththe volume of the '^"oyUnder of baae
PQ and height CO. ^
JiEx. 6. Find the volume of the solid generated by the rev f
olution about the
X-axis of the arc of the curve y = (x + l)(x+ 2) between t'che points whose
abscissas are 1, 2. L
Ex. 6. Find the volume of the cone generated by the revc 'jiution about the
X-axis of the parts of each of the following lines intercepted Ibetween the axes "
(a)2x + y = 10;^
(c) 4x - 5j/ + 3 =
-S0 ;
(6) 7x + 2y4-3 = 0; (d)3x-8y = 5. '
Ex. 7. Find the volume of the cone generated by the r /evolution about the
y-axis of the parts of each of the following lines intercepte fd between the axes "
(a) 4x + 3y = 6; (c) 5x - 7 y + 35
r|=;
(ft)3 X - 4 y = 6; ((Z)2 x + 6 y + 9 = )0.
*
Ex. 8. Find the volume of revolution about the x-axi % of the arcs of the
following curves between the assigned limits :
(a) y2 = x3, X = 0, X = 2 ; (6) (a^+ x'^)*= a ^ x = 0, x = a.
Ex. 9. Find the volume of the solid generated by tVierevolution about the
X-axis of the curve y^ = ex from the origin to the poir/those abscissa is Xi.
Ex. 10. Find the volume of the solid generated *by the revolution of the
same arc as in Ex. 9 about the y-axis.
Ex. 11. Find the volume of the solid generated by the revolution about
the y-axis of an arc of the curve in Ex. 9 from the origin to y = y^.
Ex. 12. Find the volume of the prolate spheroid generated by the revolu-ion
oi the ellipse " + ^ = 1 about the x-axis.
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30-31.] GEOMETRICAL APPLICATIONS 73
31. On the graphical representation of a definite integral. In
Arts. 4, 6, attention has been drawn to the principle that any
definite integral, whether it denotes volume, length, surface,
force, mass, work, etc., may be graphically represented by an
area.* A simple illustration may put this in a clearer light.
Ex. Find the volume of the right cone generated by revolving about the
jc-axisthe line drawn from the origin to the point (4,1).
Let P be the point (4,1),
andlet
POQbe the cone
ofrevolution. The equation of
OP is 4 2/= ".
Hence,
vol. P0"= fVy2(fx
The volume isthus | ir cubic
units of the same kind as the
linear unit employed. In
order to represent this volume graphically, draw the curve OHH whose
equation is
" x^ being the function of x under the sign of integration in (1);and draw
16
the ordinate CB at " = 4. The area BOG graphically represents the volume
.POQ.For,
area ^0C= ( ydx
Jo 16
Equations (1) and (2) show that the number of cubic units which indi-ates
the volume of POQ is the same as the number of square units which
indicates the area of BOG. In the same way, if the ordinate NH be drawn
at any point iV, for which x = a, say, it can be shown that:f\
ra^ denotes
both the number of cubic units in the cone MOL and the number of square
=r
"dx (2)
* On account of this property the process of integration was called by
Newton and the earlier writers*' the method of quadratures."
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74 INTEGRAL CALCULUS [Ch. IV.
units in the area HON. It is thus apparent that the number of cubic units
in LMPQ is the same as the number of square units in NHBC, namely
^^(Q^-a^). Hence,
vol. POQ : vol. MOL : vol. LMPQ
= area BOG: area HON : area NHB C. (3)
If the curve y"^
rmrx^ be drawn, the numbers which indicate the areas
will be wi times the numbers which indicate the volumes of the correspond-ng
sections of the cone. But the ratio of any two right sections of the cone
will be the same as the ratio of the two corresponding areas, and proportion
(3)will stillhold. The curve 2/=
A wi^^'*^ can therefore be used to represent
the volume. It is sometimes well to use a multiplier m for the sake of con-venience
in plotting the curve that will graphically represent the integral.
Note. If the firstintegral curve (seeArt. 15) of OHB, namely,
JoX2
_
_L^3-3
'o 16 48'
be drawn, its ordinates represent both the areas of the segments of OHB
and the volumes of the segments of the cone POQ measured from O.
32. Derivation of the equations of certain curves. Oftentimes,
when a curve is described by some property belonging to it,the
formal analytic statement of the property involves differential
coefficients. In these cases the derivation of the equation of the
curve
consistsin finding a
relationbetween
the coordinates which
will be free from differentials. Examples of this have been given
in Art. 12. A few additional simple instances are introduced
here. In the larger number of cases the derivation of the equa-ion
of the curve will require a greater knowledge of differential
equations than the student possesses at this stage; and hence
further
problems ofthis kind
will
be deferred
until
Chap. XIII.
Ex. 1. Determine the curve whose subtangent is n times the abscissa of
the point of contact. Find the particular curve which passes through the
point (5,4).
Let (",y)be any point on the curve. The subtangent is y ". By the
given condition,^
dx
y'^=nx.ay
This may be written, " =^.
X y
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31-32.] GEOMETRICAL APPLICATIONS 75
Integrating, log c + log x = n log y ;
whence, y" = ex. (1)
All the curves obtained by varying c, satisfy the given condition.
If one of the curves passes through the point (6,4),for instance,
4" = 6 c. (2)
Substitution in (1)of the value of c from (2)gives
5y" = 4"x,
as the equation of the particular curve through (6,4).
What curves have the given property forw = l? n = 2? n = }? w = i?
Ex. 2. Find the ctirves in which the polar subnormal is proportional to
(isk times) the sine of the vectorial angle. What particular curves paas
through the point (0,2 tt)?dr
The polar subnormal is "
. By the given condition,dd "
^= ifcsin ^.
de
Integrating, r = c " ifcos ^.
For the curve that passes through (0, 2ir), 0 = c " k; whence e = k.
Hence, the equation of the particular curve required is
r = A;(l cos^),
.theequation of the cardioid.
Ex. 3. Determine the curve in which the subtangent is n times the sub-ormal
; and find the particular curve that passes through the point (2,3).
Ex. 4. Determine the curve in which the length of the subnormal is pro-ortional
tothe square of the ordinate.
Ex. 6. Determine the curve in which the subnormal is proportional to (is
k times)the nth power of the abscissa.
Ex. 6. Find the curve in which, for any point, the length of the polar
subtangent is proportional to (isk times)the length of the radius vector.
Ex. 7. Find the curve in which the angle between the radius vector and
the tangent at any point is n times the vectorial angle. What is the curve
when w = 1 ? when n = i?
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76 INTEGRAL CALCULUS [Ch. IV.
EXAMPLES ON CHAPTER IV
1. Find the area of the figure bounded by the curve ac* +aafi
+a^
+ 6^ = 0, the "-azis, and the ordinates at as = 0, x = a.
2. Find the area inclosed by the curve scj^* y*+ 2 y* + 6 and the lines
a; = 0, y = 0, y = 1.
8. Find the area included between the parabolas y^ = 4 ox and x^ = 4 ay.
X X
4. Find the area included between the catenary y =
-(e*+ c *), the
axes of coordinates, and the line x = c,
5. In the logarithmic curve y = "f* prove that the area between the
curve, the axis of x, and any two ordinates is proportional to the difference
between the ordinates.
6. Find the area included between the curve y = " - "
* and the line
7. Find the 'area bounded by the curve y = x^-{-ax^, the ac-axis,and
(a) the ordinates at a = " a, and x = 0 ;
(6) the ordinates at x = 0, x = a.
8. Find the area inclosed by the axis of x, and the curve y = x " x*.
9. Find the entire area of the curve y^ = a^x^ " x*.
10. Find the area included between the curve
y^ (a* x^)=
a^* andits
asymptote x = a.
11. Find the entire area contained between the curve y^ (a^ x^)= a*
and itsasymptotes x = a, x = " a.
12. Find the area included by the curve x^y^ (x^ a^)= a" and its asymp-ote
X = a.
18. Find the area
ofthe loop
of
the curve
a^^= x*
(6+
x).
14. Find the total area bounded by the curve a^y^ + b^ = a'^Hhi^.
15. Find the volume of the solid generated by the revolution about th"^
X-axis of :
(a) y2"
ajS _ a;2 between the ordinates x = 1, x = 2 ;
(6) (a^ x)2y*= a^ between the curve and its asymptotes x = a, x = " a.
16. Find the volume generated by the revolution about either axis of the
hypocycloid x^ + y^ = d^.
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CHAPTER V
RATIONAL FRACTIONS
33. A rational fraction is one in which the numerator anddenominator are rational integral functions of the variables.
The fraction is proper when the degree of the numerator is
lower than that of the denominator. If the degree of the
numerator is greater than that of the denominator, division can
be carried on until the remainder is of less degree than the
denominator. Suppose that Ny D are
rationalintegral functions
of X, and that the degree of N is greater than that of D, By
division,
D^^
D
in which B is of lower degree than D ; and, therefore,
7?In order to integrate the proper fraction " it is often neces-sary
to resolve it into partial fractions. It can be shown that
any proper rationalfraction can be decomposed into
partialfractions of the types
A B Cx-hG Ex + F
x " a {x"
aY 7?-\-px-{-q {oi?-\-px -{-q)*^
in which A, B, G, O, E, F are constants, r, s positive integers,
and x^ -\-px-{-q is an expression whose factors are imaginary.
For the proof of this and the related theory, reference may be
78
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33-34.] RATIONAL FRACTIONS 79
made to works on algebra.* Here nothing more is done than
to work some examples in the principal cases that occur in
practice,t
34. Case I. When the denominator can be resolved into factors
of the firstdegree, all of which are reed and different.
7xg + 6a;-6
X
dx.x. 1. Find C^Jzl^jtllJ aj8 _ a;2 _ 6
Ondivision, x4--7x2 + 6a:-6^^^ 6(2x-l)3c8_aj2_6aj a8-a;a-6x'
6(20.-1).
"(x-3)("4-2)
Put6(2x-l)
^A^_
_C_(1)
x(x-3)(x + 2) X x-3 x + 2^^
in which ^, B, C are constants to be determined.
Clearing of fractions,
6(2x- l)=^(x-3)(x4-2) +Bx(x-h'2)-h Cx(x-3).
Since this is an identical equation, the coefficients of the same powers of
X in each member are equal. On equating the coefficients of like powers of
X, it is found that
A-{-B-{- C = Oj
-2l + 2P-3C=12,
- (M = - 6.
On solving these equations for A, B^ C, there results
^ = 1, J5 = 2, C=-3.
Therefore, after substituting these values in (1),
J x3-x*^-6x J\ X x-3 X4-2/
= ix2 + "4-logx + 21og(x-3)-31og(x
+ 2)
* See Chrystal's Algebra, Parti., Chap. VIII., Arts. 6-8.
t A few remarks on the decomposition of rational fractions are given in
Note A, Appendix.
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80 INTEGRAL CALCULUS [Ch. V.
A shorter method for calculating A, B, C, could have been employed in
the example justsolved.
Q-
'
6(2g-l)^A
B C^^'^^^
x(x-3)(x + 2) x^x^s'^x^2
is an identity, it is true for any value of x.
Clearing of fractions^
6 (2x - 1) = ^ (X - 3)(x+ 2) +-B"
(X+ 2) + Oa;(x- 3).
On letting the factor x = 0,-4=1;
on letting the factor x " 3 = 0, orx = 3, B = 2;
and on letting the factor " +-2 = 0, or x = - 2, C = - 3.
Ex 2 r ^^Ex.12, f '^^^
.^''- ^Jx-2-x-6 Jx2-4x+l
3 ri5x + l}^. Ex.13.f2a^-6x"-4x-ll^
Ex 4 fC4-3x)dx. Ex.14,fx^ + 7x" + 6x - 6^
^''* *"Jx2-3x4-2 J x2 + 2x
Ex 6 f-^. Ex.15. C--^"M^dx.Jx3-x Jx(x-i))(x + g)
Fx 6fM"l)^.
Ex.16,
f x"-3x + 3
^''- "- Jx(x2-1) .J(x-l)(x-2)
Ex 7 f Ca-5)x(toEx.17. fJ^+i)^.
E^ 8rC3x + l)dx
. Ex 18.f(8""-31x2+41x-6)dx,
^''- *"J2x2 + 3x-2 Jl2(x*-6x8+llxa-6x)
...,,|ai^. E.....jg::;f.:g[Suggestion. Assume the fraction equal to 1 + 1-etc.]
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34-36.] RATIONAL FRACTIONS 81
35. Case II. When the denominator can be resolved into linear
factorj all of which are real and some of which are repealed.
Ex. 1.r6^-8x"-4x+l^
a;8-f a;2
in which A^ B, C, /", are constants to be determined.
Clearing of fractions,
6a;8 - 8a;2 _ 4 X 4- 1 =
-4x(a;-
1)2+ B(x-
1)2+ Gx^^x-
1)+ Dx^.
Equating coefficientsof like powers,
A + C = Q,
-A+ B-C+D=-S,
A-'2B =
-4,
B = l,
On solving these equations it isfound that J[ = " 2,-B
= 1, C = 8, i" = " 5.
Therefore,
r6^8_8x2-4x+l^^r/ 2 1_8
5_\^
Ja;2(a._l)a
^
J[ x^x^^x-^l (x^iy)
=
-21ogx~i+81og(a;-l)
+ .
^
X x"1
Ex. 2. f-H^L-. Ex. 7. fi^i^Zlii^..
")(x+ l)2 J (iC-3)2
Ex. 3. f(^-^)^. Ex. 8. f(^-1)^
.
J (X-3)2 "/X8 + 2X2 + X
2(fo;
a)8
Ex. 4. f2(x"21^. E^^ 3, r_a^J (2x+l)2 J(x +
Ex. 5. ff-" 5^-W. Ex.10, fi^ii^cto.J Vx + a (X 4- 6)V J x8 - x2
Ex. 6. f 2^ Ex.11. f_i2x-_6}^_,
" (3V6 -2 -x)3 -^(aJ+ 3)cx + 1)"
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82 INTEGRAL CALCULUS [Ch. V.
Ex.12, f ^^: Ex.14. f9(2 + 4x-x")da^,
Ex.13, f" ^^" Ex.15. fi?i"il^.J(x2-2)a J
x(x-l)8
36. Case III. When the denominator contains quadratic factors,
the linear factorsof which are imaginary. This case can be sub-ivided
into the two following :
(a)When all such quadratic factors are different.
(b)When one or more of them is repeated.
The latter case seldom occurs in practice. For each power,
from the firstto the nth, of these quadratic factors, a numerator
of the form Mx-h
N, in which 3f, N are constants, should be
assumed.
Ex. 1. Fmd f
^^^.
J x8 + 4 X
x(x2+4)'~x'
x2 + 4
Assume "
i=^ +
^"^-
Clearing of fractions,
4 = J[(x2+ 4)+x(5x+ C).
Equating coefficientsof like terms,
A + Bz=Oj
C = 0,
4^1 = 4.
The solution of these equations gives
^ = 1, B =
-l,(7 = 0.
Hence,
f_4^=ffl__^UJx" + 4x JV" X2 4-4/
= logx-ilog(x2 + 4)
= log--^=.Vx2 + 4
Ex. 8. Find f ^"^^ " dx.J(x-2)(x2-2x + 3)2
Assume-^"1
"
=-^+.-g^+^
+"^* + ^
(x-2)(x2-2x + 3)2 x-2 x''^-2x + 3 (x2-2x + 3)2
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35-36] RATIONAL FRACTIONS 83
Clearing of fractions,
"8+l= ^(x2 + 2x + 3)2+(5x4- a)(x-2)(a;2-2x+3)4-(I"x+Jg?)(x-2).
Equating coefficients of like powers of x,
A + B = 0,
-4^+0-45= 1,
10^ + 75-4C4-/" = 0,
-l2A-QB-^lC-{-E-2D= 0,
9A-QC-2E=l.
The solution of these equations is, A=l, B= " 1, 0=1, i" = 1, J" = 1.
Hence,
C (x^+l)dx^r/_i
^-1+ x"l \^
J (" - 2)(x2 2x + 3)2 J\x-2 x2 - 2x + 3 (x2 2x + 3)2;
= log ^1^2 ^ r 2dx
Vi2^2^M^2(x2 - 2 X + 3) J
(a;2_2 X +
3)2
The last integral can be found by means of reduction formulae to be given
in the next chapter.
2)dx
1)2Ex.3. fi^"J:}!l?I. Ex.6. (i^"^
J X8 + X J (x2-f
Ex.4. f 2X2 + X + 3^^ j,^ ^
r (2x2+ x + 2)dxJ (x+ l)(x2+ 1) J (3x + 2)(2x2 4-4x + 4)
Ex 5f3a^+6x8-fx2+2x+2^
^^ gr(3x2 "- 17x + 33)(fx
J 3x8+18x J x8-6x2 + llx
j,^ 3ry6-3x*-22x8+17x2-23x + 20
J (a;4.4)rx2 4-l)
dx.
Ex.10. (-^+7x^4-13^.J (X2+ 3)8
E^ 11r3x" + 3x^ + 30x2+17x + 75^
J Cx2 4-6)8
Ex.12, f^i^c^x. Ex.13, f (^^-^)^Jx8-+-3x J x* 4- 6x2 4- 8
Ex. 14. ( -^^-^ dx = log-^"i-
+ i tan-i ?'.
Ja;8+ x24-4x + 4 (x + 1)^^
2
INTEGRAL CALC. " 7
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CHAPTER VI
IRRATIONAL FUNCTIONS
37. The integrals of irrational functions can be found in only
a few cases. In some instances, by means of substitutions, these
functions can be changed into equivalent functions that either
are in the list of fundamental integrals, or are rational, and
therefore integrable. In other instances the integral can be
found by means of a formula of reduction. A few irrational
forms have been discussed inpreceding articles.
38. The reciprocal substitution. Sometimes the integration of
an irrational function is facilitated by the substitution
x = \dx=^-\dt,
Ex. 1. Find {y^^^dx.
On putting x =i, f ^^^^ip^da;-- f(a^^a !)*"""
(o""g 1)*
dx
Ex. 3
x-2Va2 - x^
dx
.2.
f ^ Ex.5, r
f-^. Ex.6, f^
Ex.4. |" ^- Ex.7, f^^^cto.84
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37-40.] IRRATIONAL FUNCTIONS 85
Ex. 8. f ^ Ex.11, f ^
Ex. 9. f /^ .Ex. 12. f ^^^ " -.
"^a;Va'^ - "" "^ (2ax- x^y
Ex. 10. ("^=' Ex. 18. f:^^"^cto."^ " Va;a - a^ J x^
39. Trigonometric substitations. Although the integration of
trigonometric functions is not discussed until the next chapter,
it may be stated here that trigonometric substitutions sometimes
aid the integration of irrational functions. The following sub-titutio
may be tried :
(a) x = a sin 6 for functions that involve-\/aF"3?y
(b)x=a tan 0 for functions that involve VoM^,
(c)x = a sec 6 for functions that involve Vo^^^^^^
Ex. Find (Va^x^dx. (See Ex. 1, Art. 28.)
On assaming x = asin^, (22;=:acos^d^,
and(Va^-x^dx =
a^(co"^ede^f(1 + Co82 e)d$
2 2
40.Expressions
containingfractional
powers ofa
+6a?
only.If
71 is the least common multiple of the denominators of the powers
of a 4- hx, these expressions can be reduced to the form
F(XyVoT+bx),
If F(uyV) is a rational function of u, v, then jF(x, "y/a~+bx)dxcan be changed into a rational form by means of the substitution
a-\-bx
= ^.
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86 INTEGRAL CALCULUS [Ch. VI.
For then x = ^"^, dx=:^^dz,0 0
and hence, Cf{x,VoT"S) d^ = ^iJV/'?lzi^,\^-'z.
Expressions that contain fractional powers of x only, belong to
this class. This is apparent on putting a = 0 in a-^- bx. If n is
the least common multiple of the denominators of the exponents
of X, the function can be changed into a rational form by the
substitution
x=z\
Ex. 1. Find f ^
"^ x^/a^ + bx
On making the substitution a^-\-
bx = z^, x =
^ ~ ^
,dx =
"dz.b b
Cdx
_ Q fdz
a z-\-
a
=
^log
^^'^-^bx-a
" Va2 + bx^-h a
.2. Find C~^^"d
"^1 + a* .
The L. C. M. of the denominators of the exponents is 6. If x = "*, then
dx=:6z^dz, and
= 6 x*(fX - io;*+ iX*- 1)+ 6 tan-i
x*.
Ex. 8. Find (-y^dx. Ex. 4. Findf 3 + 5(x^ + x*)dx^
-^* - 1
-^ 15(x + X*)
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40-41.] IRRATIONAL FUNCTIONS 87
f"i+V^
,Ex. 9. f__"
__i5 ;
Ex. 6./"(a;i-2x*)(te Ex.10. C2^EI1"1
dx.
j,^^ ^^V^-7-^+12v^^^ Ex. 11. J(3-x)V(F^r^
x(V^~\/x)
/^,
Ex.12, f ^^^.
Ex.8. lxVa+6x(Zx.'^(2x + 3)*
41. Functions of the form /{a?*,(a + fta;^)} . a?cfa;, in which w,
n are integers. If f(u,v)is a rational function of u, v, these
functions can be rationalized by means of the substitution
a 4- 6a^ = "**.
For thea, 2bxdx=nz''~^dz, x^= "
^^^,and the function becomes
0
Ex. 1. Find f ^^(See Ex. 10, Art. 38.)
This belongs to the form above, since"
X Vx2 - a^ x2 Vx2 - a2
On putting x^ " a'^ = z'^, xdx = z dz^ x'^ = z^-\-a\ and
Jdx _
C_dz_
=lton-iia a
a a a X
Ex. 2. f-^-^"- (SeeEx. 8, Art. 38.) Ex. 8. f ^^^.
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88 INTEGRAL CALCULUS [Ch. VI.
Ex.4, r ^^ Ex.6, f ^^
Ex. 6. If /(", v)is a rational function of u, v, show that
4- ^'f^a'^can be rationalized by means of the substitution
^^"^ = """.
ex +d"
42. Functions of the form Fix, y^x^ + aa? + b)dx9 Fiu, v)
being a rational function of u, v. If the radical be Vma^+|M5+g,
it can be written Vm \a^-{-"x-^-"'
m m
The given function can be rationalized by assuming that
Va?^4- aa;-f
6 = " " a?,
and then changing the variable from a; to a;.
For, squaring and solving for ",
a + 22;
whence, " " a; = " -f "gH- o
and ,^ = 2^!"_^"^d..(a+ 20)*
Therefore, on substitution,
Ex." ^^
x\/a;2+ aj + 1
Assume vj^H-lc + l = " " as.
Squaring and solving for x, x =-=
i-.
1 4" 2 ^
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41-43.] IHBATIONAL FUNCTIONS 89
Hence, ^^2(.^+ . + l)
(1+22)2'
and Vga + g; + 1 = g - x =
^^ + ^ + ^"
1 + 2"
On substitution,f
"
^= 2 f
-^^"^ " Vx2 + X + 1 ^ "2 _ 1
^.2?+l
_ iQg g-l+Va;2 + y + l
" + 1+ Vx2 + X + 1
43. Functions of the form /(", V-x^
"\-ax-\-6) da?, /(w, v)
being a rational function of w, v. If the radical be V"-mar^4-i"aj-f,
it may be written Vm\/-a^" " + -
^^ *^" factors of
" a? '\-aX'\-bare imaginary, V" aj2 4- aaj + ftis imaginary. For,
if one of the factors isx"a-^-i^,he other must be
"("""""/?),
and hence,
" a^H-aaj-|-6 = " (" " " + */?)("a "
t/?)
which is negative, and has an imaginary square root whatever x
may be. Only cases in which the factors of " a?^-f-
aa; 4- 6 are
real will be consideredhere.
Let"
0^4- "" + "=(""
a)(fi"
x).
Assume "\/"3i^-\-ax-fb
or V(a? a)(fi"
x)= (x "
a)z.
Then, squaring, /?" ar = (a? a)^*,
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90 INTEGRAL CALCULUS [Ch. VI.
and V-a^ + aa;4-" = (a? a)2; = %^^ "
1 +2r
Hence, on making the substitutions,
which is rational, and accordingly integrable.
Equally well, the substitution, V(aj"
a)(/? x)= (fi x)z,
might have been made.
It follows from this article and the preceding article that, if X
is the indicated square root of an expression of the second degree
in X, every rational function / (a?,) is integrable.
Ex. 1. Find (- ^
Assume
V- x2 + 5 X - 6 = V(x - 2)(3 _
x) = (x - 2)2.
From this, on squaring, 3 " x = (x " 2)z'^,
Hence, ^ = '-^^
dx =^^^^,
and V-x2 + 6x-6=(x-2)g="^
"
'
Therefore, on substitution,
r dx_
_ 2C dz
Ja^V-x2+ 6x-6
J222 + 3
=
-V|tan-iV|"'
^3 "3 (a; 2)
Ex. 8. f ^ Ex. 8. f- *"
"' (1 + x2)\/n^ "' V2x2+3a; + 4
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43-44.] IRRATIONAL FUNCTIONS 91
dx
Ex.4. (y!A^^"J^dx. Ex.6, r
Ex.6, f^ + ^
(to. Ex.7, r^^
xWl- x^
V6x-x^dx.
44. Particular functions involving "y/aix'^ftoc+ c.
(a)If a is positive,
r ^^
=-l=log(2aa;" + 2Va Vaaj^
-j-
fta?+
c)JVaa^ + 6x.+ c Va(Ex.43, Chap. III.)
(")If a is negative, say " aj,
r ^-^=JL
sin-i
^"i"^-^_.(Ex.44, Chap. III.)
(c)r (-4^tj)_^..
"/ V aar^ H- 6aj-f
c
Since " (cux^-{-
bx-{-c)
= 2 ax-\-
b,
dx
and Ax-\-B=:^(2aX'^b)-^B-4^,a 2a
"/ Vaar^ 4- fto?-f
c^ ^*^ Va^
"i-bX'{-c
\ 2a)J -y
dx
A
= " Vox"^-h
6a;+ c-h
an integral of the
a
form (a)or (6)above.
_, , ,,. ^ r (x + 3)(toEx. 1. Find i "
^^ ^^ "
J\/x2 + 2a; + 3
Since " (x2-j.2x + 3) =2a; + 2, and x + 3 = i(2x + 2) +2,
dx
C (x4-3)fte
_l.r
(2x + 2)dx C dx_
^ Vx2 + 2x + 3
~
2 J Vx-^+ 2x + 3 ^ V{x + 1)2+ 2
= Vx-^+ 2x + 3 + 21og(x+ 1 +"/x2 + 2x + 3).
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92 INTEGRAL CALCULUS [Ch.VI.
Ex.3.)dx
"^ V5xa~2a;H-7 -^ V6-3x-2x2
Ex. 8. r^^
"
Ex. 7. (J'^"^dx,
Ex.4, f ^Ex.8. fi^xjLli^,
Ex.6, f ^^Ex.9, f 7^ + ^
dx.
^V-2x2 + 3x + 4 -^ V3x^-3x+l
(d) f^ r
" ^"^="te,^ (x" a)
Voa^ + 6j;4- c^ {x "
a)Va^^f6"^fc
JIf,-^
being constants.
On putting a? " a =-,
da? = "
-^dz,andthe firstintegral takes
the form
-/;dz
in which A, B, (7, are constants. The first integral is thus
reducible to (a)or (h).
Since Mx-f--ZV=
3f(a; a)+N+ Ma,
.
r {Mx^N) ^^ rM{x^a) + N-^Ma ^^
"^ (x" ")Vaic^4-bx + c
^ (x "
a)Vao^ -\-bx-\-c
= mC^
4-(N4-Ma)C^
"^ Vax^ 4- 6a;-h
c "^ (a? a)VaJ^ + 6a?4- c
The two integrals in the second member have been considered
above.
Ex. 10. Find^ ^*
l)Vx^^"=^2x-f3
On putting x " 1 =-,
dx = "
-dz,and
z g2
f ^y-
=-f^^
=:-.J--log(gv^+v/rf2i^)"^ (x-l)Vx2-2x + 3
"^Vl + 2^2 V2
=Jl log
Va^'-2x + 3+V^_
V2 "-!
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44-46.] IRRATIONAL FUNCTIONS 93
Ex. 11. Find ( {^^-^^)dx
"^ (aj- 2)V2x2 + 8x + 10
Since i^-"l=3 i-, and 2x2+ 8x + 10 = 2{(x + 2)2+ 1},
dx
(x+2)V2x2+8x + 10 V2'^"/(x+2)2+l -^ (x+2) V(x+2)'H1
The integrals in the second member belong to forms already considered.
Integration gives the result,
Alog(x + 2+Vx2 + 4x + 6)+2v^log^^' + ^^ + ^"i.V2 aj + 2
Ex.12,f (x-+ x + l)dx (suggestion:"^=x+4+.^.\"^ (x--3)"/5x'*-2(5x+34 \ x-S x-S J
Ex.18, f ^
"^ xVx^ + X + 1
Ex.14, f (2xi^l^^^.'^(x-l)Vl + 2x-x2
45. Integration of x'^(a-\-bx^)Pdx: (a) by the method of
undetermined coefficients; (b) by means of reduction formulae.
Some integrable functions are of the typex'^(a bafy,in which
a,b, m, n,
jp,are
constants.The
exponentn can
always be
positive. If the integration of an expression having this form
is possible, it can always be effected by the method of "inte-ration
by parts." A shorter method, however, may sometimes
be employed. The given integral is expressed in terms of a
function of x not affected by a sign of integration, and of
another
integral
which
is easier to integrate than theoriginal
function.
(a)EuLE. Put IQif^(aboifydxequal
to a constant times
one of the integrals
Car-''(a- bof^yx, jV+"(a-|-bx^ydXy
Car(a-\'^baf)^^dx,
Car(a+ baf^y^x.
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94 INTEGRAL CALCULUS [Ch. VI.
plus a constant times x^+^(af "a?")'*+^,n which A, fi are the lowest
indices of x and of (a4- bx^)in the two expressions under the inte-ration
signs. Then determine the values of the constant coeffi-ients
thus introduced.
For example, on taking the firstof the four integrals referred
to in the rule,
Cx^(a-\-bafydx=Aar-''+\a-\-bo^y-^^i-B Car-^a-^-baf^ydx.(1)
Here A, ft, the lowest indices of x, (a-f 6af),under the sign
I,are m " n, p-, and from this the term
.4ar~"+^(a|-bx^y'^ s
derived. In order to determine the coefficients A, B, differentiate
both members of equation (1). The result, after simplifying, is
x"" =Ab (m -\-np -\-l)x -{ Aa(m
" n-f 1)-f
5.
From this, on equating the coefficientspf like powers of x,
j^^1
^^ a(m-n 4-1)
6(m 4- np -h 1)'" (m 4- np 4- 1)'
The substitution of these values in (1)gives
Similarly, by connecting iirf"(a- bif^y
x with the other inte-rals
mentioned in the rule, the following results are obtained :
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45.] IRRATIONAL FUNCTIONS 95
/
+
an(p + l) J"''"("""'")^^^"fa^ [D]
In each of the four integrals with which jic"(a- bafydx may
be connected by the rule, m is either increased or diminished by
w, or else p is either increased or diminished by unity. The values
of m and p will indicate theone
of thefour
which is simpler than
Iaf"(a-j-hx"")"Xj and may preferably be connected with it. Suc-essive
applications of the rule are necessary in some cases.
(b)The results A, B, C, D, can be used as formulae of reduc-ion.
It is necessary only to make carefully the proper substitu-ions
for m, w,
p,
in them. It is notnecessary
to
memorizethese
formulae, as they can be kept for reference.
It is well to be familiar with the process of deriving A, B,
C, D, so that they can be readily obtained when required*
if necessary. Formulae A, C fail when m-h w/? + 1 = 0, B fails
when 771-f
1 = 0, and D fails when p -f-1 = 0. In these cases
other methods can be used.
Ex. 1. Find f ^^^.
Here, w = 4, n = 2, p = " J, a = a:^, 6 = " 1.
On connecting this integral with I "
^:^^^^^by the rule in (a),it follows
." -^ a//z2 x^that
* Another method of deriving these formulae, that of integration by parts,
is given in Note B, Appendix. Other formulae of reduction can be obtained
by connecting
\ x'^ (a-{"bx")Pdx vf'ithfx^-^-^Ca6a;'")''+*dxnd
(x'^+''-^(ahr')P-^dx
in the manner described in therule
in
(a).The
student may
derive the
formulae in this case as an exercis'fe. (See Edwards' Integral Calculus,
Art. 82.)
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96 INTEGRAL CALCULUS [Ch. VI.
(1) C.^^=AxW"^^^-i-B^C^^^.
" thus,
(2) r x^dx^ ^ix Va^TT^ + Ci f
"/a2- a;2
^ Va^ - x*
Hence, on substitution in (1),
(3)C x*dx
=A^y/ai_^2 + BxVa^^^^ + C (^ -
"^ Va2 _ a;^ -^ Va-"-
x*
On differentiating and simplifying,
X* = 3^x2(a2 -
x2)-Jx* + B(aa- x2)-
Bx^ + C.
On equating coefficients of like powers and solving for A, B, 0, it is found
that
A =
-i,B =
-la^C=ia*.
Substitution of these values in (2) gives, since f = sin-i -"
f" ^^=r = - 13 a* sin-i?
-
x(2 x2 + 3a2)
Va23^2 \.
The coefficients A, Bi, might have been determined in (1),and ^i, Ci,
might have been determined in (2).
The integral could also have been found by the application of formula [A]
twice in succession.
Ex. 2. Find T"
^
Here m = 0, n = 2, p = " k, a = a^, 6 = 1.
Connecting the integral withf (x*-f a^)^^+^dx,
(1) f(x2+ rt2)kax = Axix^ + a2)-"+i+ b((x^+ a2)
"+idx.
On differentiating and dividing the resulting equation through by
(x2-a2)-*,l = ^(x2 + a2) + 2 Ax^(l - ifc) B(x2 + a2).
On equating coefficients of like powers, and solving for A, B, it is found
that
2o2(*-l) 2a^{k-l)
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45.] IRRATIONAL FUNCTIONS 97
Substitution of these values in (1) gives
J (xa+ a2)*
"^
2 a2 (ifc 1)I(x^+ a2)*-i"^ ^^
* ~
^^J (x^+ a2)*-i*{This result might have been obtained by the application of formula [D].
Ex. 8. (Va^-x^dx.(SeeEx. 1, Art. 23, Ex. Art. 39.)
Ex.4.C ^^
.
lV2ax-x^ = x^(2a-x)^.]^ V2 ax - xa
Ex.6. f_*!^_. Ex.6, f ^.
Ex.7, f /^ .
"^ Va2 - xa -^ (^2 _
a.2)f'^ x8 Va^ - x^
Ex. 8. fX V2 ax - x^dx, [Suggestion. Put fx^(2a -
x)^dx
= Ax\2-
x)i + J?x*(2 -
x)* + Cx*(2a -
x)* + D(x~i(2a -
x)"*(te.]
Ex.9, f ^.
Ex.10. C ^'^^.
"^x*Vaa-xa "^(xa+ a^)*
Ex. 11. Show that
Jx"*(ixx'*-^ yga -- x^
. (w - l)aaC x^'dx
Va2 - x2"" t" J
Va^-aja
Ex. 12. Show that
fX- V^^:i?(to =^^^^^^^"^^^
+-ili~
f ^^^.
J w + 2 w + 2 J
-v/^nr^
Ex. 18. Show that
dx Va^ - x2, n-2 r dxdx
____
Vqa - x^ n-2 T
" x"A//i2_a-2(n - l)a2x"-i (n - l)a^J lx"Va2_xa (" - l)a2x"-i (n - l)o2j a?"-2 Va^ - x^
Ex. 14. Show that
r x"*dx__
x""-W2ax-xa (2m-l)qr x"* 'dx
"^V2 ox - x2*" m J
V2ax-x^
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98 IN TEG HAL CALCULUS [Ch. VI.
MISCELLANEOUS EXAMPLES.
-Vxdx 15. f ^
r dx 16.r (a;-4)dx
8f____J^^?____. 17. f ^^
, ,/c",.o^+" 1 "ito partial frac-\
4rv/i^ + Vx-6^ (SeparateJ"
^^^j
"^ (X "
a)Vx " 6
"^V2x6 + 3x8 + 1
e. I '^^
xdx
'(x2-2)vGcir33
)dx
V-3x2-6x-2
dx
\/-27+ 10x+5x2
8.f V-^^ + ^^^-dx.
"^ (mx "
n)Vwix + n
22. f ^
9rV2x-x2(V2x-xHflr) + /t^^
'
J 34x-17x2 28. r
^^r(x''^ 3x+5)dx^
J(X + l)Vx2+l
20r (x- + 2x + 3)dx
" (x- + 2 X -" 3)Vl - a;-2
21. JV6H^"2(VSH"2x2+ax)2dx.
,2Va2-x2
x^dx
Va2 _ x3
10.fV2x-3(V2-3x+V2x+3)^^
"^ (2x-3)\/6-6x-6x224. J
11.finl^ili)^
^^r
^ax
"^ Vx2 + 2x + 426. J-p==:
Va2 - x2
12.r(^ + l)V^-2dx.
26. rxVa2
" x2
13. J(x-l)^(xiridx. 27. J^^'-^'cZx.
14. f_ (fX
i)\/x2TT
28. j'x2Va2-x2(ix.
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45.] IRRATIONAL FUNCTIONS 99
J iK" va X ax. J ^^_ JJ2^^ _^
.
331 dx
86.* *^*
1)"
dx.80. f
'
'^ "
45CVl + x + x^
81. I.f
"
^ f V^"fa
47. f(a-''+6*-*'')V(a2-x")(a:'-6"
f- ^
(""" o")* 48. ^^V2ax-3fiers-" ?
(te.
84. j-(x""a")tcte.^^
r-x" + :^ +.3x
+ 9.
*
Jx" + 9x*+27x2 + 27'
85. f g"^
60. i "'-^^ dx.r xg + 6
^ Vx2+
!
\Vx2+2 VxM^ /
61. f ^^ + 2x+l^,
88. f ^^ "^V4x2 + 4x + 3
"
Jv2^^3:i^
^^r
x^+x-fj^^^89. Jv^^^^T^dx. "^Vx2 + 2x + 3
do f_^_ *^- f.
^'-^-3dx.
*^- J(x2+ aa)s-
"'V-3 + 12X-9X2
41f ^^ 64. r Vlx^ + mx + n dx, I negative.J (X*+ a*)2' -^
4Sf a;2dx
66. fJ(x8-a")2* -^(xa+a-
dx
(x8 a")2*-^ ("*+ o^)
Vxa - a^
(x+l)dx43. f ^
66. f-J (x2 2 X + 3)2 -^ (x2-2x + 3)2 -^ (2x2-2x+J)V3x"-2x+l
INTEGRAL CALC. " 8
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CHAPTER VII
INTEGRATION OF TRIGONOMETRIC AND EXPONEN-IAL
FUNCTIONS
Integration by parts and the use of substitutions will be found
very helpful in obtaining the integrals of trigonometric and
exponential functions.
46. Isin* xdai"9 I eo8**a?"la?9 n being an integer.
(a)If w be a positive odd integer, say 2m + 1,
Isin"a;cto= jsin**+^a;daj= jsin** a? sinxdx
= " I(1"
cos'o;)"'(cosx).
The binomial in the latter form can be expanded, and the inte-ral
found term by term.
Ex. 1. Isin*xdx = " f(l--os* x)d(co8x) = - cos aj + J cos* " + c.
Similarly,
jcoa^'^+^xdx= I(1" sin*")""(:?sina?) sin a?"
^ sin*a? +""".
Ex. 3. (cos^xdx=
({l-am^xyd(8inx)= ((1-2 Bm^x-\'8m*x)d(Bmx)
= sin 05 " }sin"x+ 1 sin* a; + c.
(b)If n be any positive integer, integrate sin^acZa? by parts,
putting
w = sin'*~^aj,d'y= sinajdaj.
100
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46.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 101
Then, du= (n " 1)sin**^ x cos xdx, v = " cos x ;
and Isin'^xdx=
"
sin**"^ajcos a? + (n"
1)jsin"~^a?cos*ajda?
= " sin*-^a?cos x+(n" l)(sin^-^a?(1 sin*a?)daj
= " sin**~^a;cosa?-|-n " 1)|sin*"^a?dic
" (n"1) Isin^ajda?.
By transposing the last term to the first member, combining,
and dividing through the equation by w, the result is
fsin'xdx = - si^-'^cosx^^L^Jsi^n-.^^.A]
The result A can be used as a reditction formula. Successive
applications of it leads to- I da? or I sin a;da? according as w is
even or odd.
Ex. 3. jsm^xdx=-.^^^^^^ysmxdx
1 2= " - sin^o; cos x " cosas + c.
This result may be compared with that of Ex. 1.
On integrating cos"a?c?a? by parts, putting u = cos**"^a?, dv =
Gosxdx, the following reduction formula is obtained,
/'^
, sina;cos'*"^a? n " l r^
" _ r-Ri
cos" xdx = "
I cos**~^ x dx. Ii^Jn n J
The deduction of B is left as an exercise.
(c)Suppose that n is a negative integer.
The value of jsin"^
a; (fa?in A is
J n " 1 n " lJ
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102 INTEGRAL CALCULUS [Ch.VII. 46-
On changing n into n-|-
2 this becomes
/.-,
, sin""*"*cos a?,
n
-f
2 /* .
". 2 ^ r^i
sin" xdx=^
'-"
I sin'*^* x dx, fC |
This can be used as a formula of reduction when n is a nega-ive
integer.
Ex. 4. f^^
= f(sinx)-8 da; = - i cos x(8in")"* + f(sinx)-i dx
cos a;,
r^""^j^= 7"
"
h I CSC X dx
1 X
= " - cot as CSC as + log tan - + c.
2 2
Similarly, on solving for Jcos^'^ajcfa?in B, and then changing
n into n-H
2, there results,
fcos'xdx-'J^^^^^
n"4fcos-^^xdx.D]J n + 1 n + lJ'--'
This is a formula of reduction for jcos"a?(ir when w is negative.
It is advisable to remember the method of deriving A, B, C, D,
so that they may be readily obtained when necessary.
Ex. 6. (a) icos^xdx, (6) rsm*a;dx,(c) ism^xdx,
Ex. 6. (a) rsin*a;da5,(6) isin^xdXj (c) Tsin^ajcte.
Ex. 7. (a) (cos*; da;, (6) Icos'^ajda;, (c) (cos^xdx.
(*)JJ-.'^"Jife'^^L-fe-
Ex. 9. Show that f^sin2""(to^ '^'^""^^^~ ^^
.^.Jo 2.4.6...2W 2
IT
Ex, 10. Showthatpsing""+^a;(fa;=^'^'^-^m
.
dx
cos^x
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48.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 103
47. Algebraic transformations. The trigonometric integral
Isin^ajc^ojan be put into an algebraic form. For, if
sin x = z,
then cos xdx = dz,
and da: =
-^^-
^^-
cos a? VI " 2*
and hence, (sin"ajcto = (^ ^
"
The second member has a form which has been discussed in
Chapter VI.
If the substitution cos a? = 2
be made, |sin" xdx = " (1"
"*)* da;.
This form can be integrated by methods already explained.
These substitutions may also be employed in the case of 1 cos" x dx,
Ex. 1. rsin*a;cfcc.(Compare with Ex. 3, Art. 46.)
On putting
" = sinx, isin"a;(a;= I - = Vl - "2 + c
= " J sin* X cos " " } cos x + c.
Ex. 2. Solve Exs. 2, 4, 6
(a),7
(a),8
(6)ofArt. 46 by
algebraic sub-titution
4a Jsec" a? dac, Jcosec"" x dx.
(a) Since sec a? =, and cosec x =
-:"
,I sec" x dx and
/cos
a; sinx J
cosec" a? da; can be reduced to the forms considered in Arts.
46, 47.
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104 INTEGRAL CALCULUS [Ch. VII.
Ex.1, (a) fsec*r (te, (6) fcosec^xdx,c) f-^, (d) f" ^"
[SeeExs, 8 (c),8 (^r), (d),6 (a),(Art.46).]
Ex. 8. Find f^
^
, ^"assuming x = atan ^. (See Art. 39, and Ex. 2,
Art. 46.)
(b) If n is an even positive integer, another method may be
employed.
Since sec* a? = 1 + tan* x, and d (tanx)
= sec' x dx,
|sec"ajdaj=sec"~*a;sec*a;daj
=r(l+ tan"aj)^~dtanx).
The binomial under the sign of integration can be expanded inn " 2
a finite number of terms since n is even, and accordingly"
^"
is an integer.
In a similar way it can be shown that
Icosec"ajcfaj = " |(14-cot*aj)d(cotx).
Ex. 3. f8ec" X dx = r(1 + tan^ x)2(i(tan);
= tan X + } tan* x + J tan^ x-f
*
c.
(Compare Ex. 8 ("),Art. 46.^
Ex. 4.' (a) isec*xdx; (6) lcosec*X(ix;
(c) i cosec" X dx ; ("i)i sec*?
(?x.
[Compare the results with those of Exs. 8 (c),8 (/), 8 (g).Art. 46.]
(c)If n is any positive integer greater than 2, the method of
integration by parts can be used.
On putting sec"
*
a? = w, sec* xdx = dv,
it follows that du = (n " 2)sec*** a? tan xdx^ v = tan x.
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48.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 106
Hence, jsec" xdx= jsec**"^ x sec^ x dx
= tan X sec"'^ a? " (n" 2)jsec**~^ x tan^ a? da?.
On substituting sec* a? " 1 for tan* a? in the last term, and solv-ng
for Isec** X dx, there is obtained
fsec-;tdx*5EL^^5^ fsec-'a^da..A]J n " 1 n " lJ
This formula of reduction leads to |sec a? cto?when n is odd,
and to Idx when n is even.
Similarly, integration by parts will give
/-, cota?cosec'*~*aj ,
n " 2 /*"
-j rT"n
cosec" xdx = 1 I cosec"*x dx, [B]
n " l n " 1*/
which, on repeated applications, leads to |cosecajda? or to I dx,
according as /i is odd or even.
Ex. 6. (a) Tsecxdx; (b) isec^ xdx; (c) I cosec^ x dx ;
(d) leasee^ xdx; (e) isec^^xdx,
[Compare the results with those of Exs. 8 (6),8 (d),4, 8 (a),Art. 46.]
Ex. 6. (a) laec^xdx; (6) fcosec*a;cto.[Compare Ex. 4(a), (6).]
(d)Transformation to an algebraic form.
dzIf tan a? = 2!, a? = tan"^2!, dx = -
-, sec*
and hence, I sec" a? da; = j(1+ 2*)*
(fe.
Also, if sec a; = z,
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106 INTEGRAL CALCULUS [Ch.VII. 48-
"
dzit follows that dx =
and hence, I sec* xdx=:( ^
dz.
In like manner, the substitutions z = cot x, z = cosec x, will re-duce
I cosec" a; c?a;to an algebraic form.
Ex. 7. Solve Exs. 3 (a),4 (a),4 (6),1 (d)above, by algebraic substitution.
49. Itan** x dx^ i cot^ x dx.
(a) Let n be a positive integer.
Then,Itan** a; cfo; |tan""* x tan* a? da?
= Itan*~* a? (sec*? " 1)cto
= Itan"*a? d (tan")
" jtan"~*a?daj
^ tan;^ ftan"-*; da;. [A]n " 1 J
This reduction formula leads to idxoY to jtanajc^, accord-ng
as n is even or odd.
N
Ex. 1. (tan*xdx=z
Ttanx (sec^
"
1)cZx=
TtanasJftanx)"
Ttanx dx
= J tan2 X " log sec x-}-
c.
In like manner,
Icot" a; da? = j cot"~* x cot* a? da; = I cot"~* x (cosec*; " 1)da;
= -
52^^fcot"-*dx, [B]
n " L J
another reduction formula.
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60.] TRIGONOkETRlC AND EXPONENTIAL FUNCTIONS 107
(b) If n is a negative integer, say " m,
Itan"a?da?= " cot^ojcto?,and lcoVxdx= (tan'^xdx.
Hence, this case reduces to the preceding.
Ex. 2. (a) itAii^xdx, (6) \ta,n^xdx, (c) itaji^xdx,(d) (cot^xdx,
(6) icot^xdx, (/) (coi^xdx:
(c)Transformation to an algebraic form.
dzIf tan a; = 2, dx =
:^""
-',
and hence, " tan**a?da? =
ij^'dz+ z'
Again, if sec a? = a;, da?^
z-yjz^1
'
n-2
and hence, I tan" xdx=^ i " dz.
Similarly, I cot^ajc^a?can be changed into algebraic forms by the
substitutions,
z = cot X, and z = cosec a?.
Ex. 3. Solve Ex. 1, 2(a),2(d) above, by algebraic substitution.
50. jsin***a? cos" ficdJaJ.
(a) When either m or n is a positive odd integer, no matter
what the other may be, the form sin"*x cos" x dx can be integrated
very easily. For, it is then reducible to a sum of integrals of
the form jsin^ a? d (sina;),r of the form I cos* a? c?(cosa;).This
is illustrated by the following example.
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108 INTEGRAL CALCULUS*
[Ch. Vn. 50-
Ex. 1. sin' X cos" xdx= TsinT (1 " sio*x) d (sinx)
=
^ sin
7x "
^ sin^ x+c.
Ex. 2. fsin*xcoe^xdx. Ex. 4. fJ^BiBLcfe.sin* cos* X dx. Ex. 4. i
/C08X
Ex. 8. fcos*X sin* x dx. Ex. 6. f?^^ "to.
"^ -^v'snri
(b) When m + n is a negative even integer, say ^2p,
sin* X cos" a; da? can be integrated by means of the substitution
tan x = L
If tana; = ^, da? = -"
r, sina? = "
.cosa; = " :
! + "" Vi+T* VTT?
and hence, | sin- a; cos** a? cto? ) ^^^^^-= jr (1+ ^'"^d^.
The last form is readily integrable. Sometimes the substitu-ion
cot x=t is better than the substitution tan a? = ^ for obtain-ng
a simple integrable form.
rsing X^
J C08*X
Iftanx = f, (^^dx=(^il + t^)dt ifi+ ifi+ e
J COfi^X ^
= J tan* X + i tan* x + c.
The actual substitution of the new variable may often be conveniently
omitted ; for instance,
f!i5l^"2x=ftan2xsec*x"te=tan^ x (1 -htan^ x)d(tanx)
J cosP X J J
= \ tan* x-\-\tan* x -f-c.
Ex. 6. f^i^dx. Ex.7. f^(te.J cos'' X J sm" X
(c)Transformation to an algebraic form.
dzIf sinaj = 2;, cos a? = vT^-"^, dx"
^
and hence, J sin" x cos" a; da?=1 2"'(1 i^^dz.
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61.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 109
In like manner the substitution z = cos x will give
J sin^ajcos^adajrs J 2J*(1"2;^*
dz.
The substitution tan x = t leads to a simple form when m + n
is a negative even integer. This has been shown above. See
Ex. 6.
51. Integration of sin^ x cos^ x dac i (a) by the method of
undetermined coefficients; (b)by means of reduction formulae.
(a) Rule. Put jsin*"a? cos** a; da? equal to a constant times
one of the four integrals,
jsin"*~^X cos" X dx, isin*"x cos**"^ x dXy.
Isin""^^ cos** a?da?, |sin"*a? cos**"*"^ a?da?,
p?ws a constant times sin''^^ cos*"*"' a?, in which p, q are the lowest
indices of sin x and cos x in the two expressions under the inte-ration
sign. Then determine the values of the two constant
coefficients by differentiating, simplifying, and equating coeffi-ients
of like terms. For instance, using the first of the four
integrals referred to in the rule,
Isin"*X cos** xdx= A sin""^ x cos***"^ x + B I sin*"""^cos" xdx.
Here the lowest indices of sin x, cos x, under the sign I,
are
m"
2, n; and from them the term ^ sin"*"'a; cos""*"^ a; is formed
by the rule. On finding the differential coefficientsof both mem-bers
of this equation, there is obtained
sin**X cos** a; = (m " 1)^ sin*"^
x cos**"*"^a; " (n-f-1) sin**x cos** a?
+ B sin"*~^X cos** x.
Divisionof
both terms bysin"*"^
a? cos** a;
gives
sin*a?= (m " 1)-4(1 sin*a?) (n-f-1)-4 sin^a?-f-
5.
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110 INTEGRAL CALCULUS [Ch.VIL
On equating the coefficients of like terms in both members^
" (m-I-n)-4
= 1,
(m-l)^ +-B
= 0;
whence, -4=
,B = "
Hence,
J,^ ^ ,
8in*^-*a5COB**+*a5fn + n
+ ^^=if8ln"~-"? C08~ 05 da?. [A]
In a similar way, by connecting I sin*" cos* a? c2a?with the other
integrals mentioned in the rule, the following reduction formulae
are obtained:
ssln*^ X COB" xdx =sin*"
+* a? 008**+ *
05
m + 1
^w + n
-f2 rsinm^aa.co8"a? da?, [B]
ysin"* 05 cos" xdx =
sin"*"*" 05 cos**~^ 05
+_WLZil.fsinmajcoS"-2o5claj.
[C]
J'in"* 05 cos" xdx = -
sln"*+ "
05 cos" +* 05
n + 1
+
^^i^^fsin"*ajcos"+"arcla5.D]
n + 1 ^
In each of the four integrals with which I sin^ajcos^ajc^ may
be connected, by the rule, m or n is increased or diminished by 2.
The numerical values of m, n will indicate the one of the four
which is simpler than |sin**; cos" a: c?a;,and with which it may
preferably be connected. A succession of steps like A, B, C, D,
may be necessary.
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51.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 111
In solving the problems of this and the following article,it
may happen that the results will not agree in form with those
given in the answers. An agreement can be made by using the
trigonometric relations, sin^ a;-f
cos^ a? = 1, sec^a?= 1-h
tan^a:, etc.
Any apparent difference in results will be due to a difference in
the methods of working the examples.
Ex. 1. (sin^ z cos^ x dx.
Assume Jsiu^ a; cos^ xdx = Asmx cos^ x-\- Bi cos^ x dx.
Differentiating, sin^ x cos^ x = A cos* x "
^iA sin^ x cos^ x-^ B cos^ a,
whence, dividing by cos^ ", sin^ x = A(l " sin^ x)" ZA sin^ x-^ B,
Equating coefficients of like terms,.
-4A= l,
A + B = 0.
On solving these equations, A = " ^, 5 = J.
Hence, jsin^ x cos^ xdx = " Isinx cos^ ^c + } i cos^ x dx ;
from this, by Art. 46, = " J sin;c cos^ " + j (sinx cosx + x)+ c.
Equally well, I sin* a; cos^ " dx might have been connected with Jsin^xt^x.
Also, 1 " cos^x might have been substituted for sin^x, or 1 " sin^x for cos^x,
and the integral found by the method of Art. 46.
Ex.2. rsin*xcos2x(te. Ex.4. C^^dx.J J sm* X
,.8. (*"-^"
" Ex.6. f5J sm* X cos* X ^8
Ex.3. 1 "^ Ex.6. \^^dx.' 'sms X
Ex. 6. Solve some of these examples by reducing them to an algebraic
form, as described in Art. 60 (c).
(b)The results A, B, C, D can be used as formulae of reduc-ion.
It is necessary only to substitute in them the proper values
for m and n. It is not necessary to memorize these formulae.
The student should make himself familiar with the process of
deriving these formulae, so that he can readily obtain them when
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112 INTEGRAL CALCULUS [Ch.VII. 61-
required* The formulae A, B, C, D of Art. 46 are special cases
of A, B, C, D above. This will be apparent on putting m and n in
turn equal to zero in the latter formulae. Moreover, jtan" a?da?,
jcot" a? da?, discussed in Art. 49, may be put in the forms
Isin" a? COS"" a? daj, I cos" a? sin"" a;da?,and solved by the methods
of this article. For the sake of practice in making the substitu-ions
a fewexamples may
besolved
by means
ofthe formulae.
Ex. 7. fsm"a;co8*"daj.
By^, fsinea;co8*ajda;-^^^^^?^^^
Af8m*xco8*x"to;J 10 lOJ
by^, f8in*xco8*a;da;-5y^i^^2^
+ |f8m2a;cos*x(te;
by^, j'sm"aico8"a;(fa-^"''""^'' + lj'co8"x(te;
by a, j'co8"xdie^M^ + |j'oo8"*"fe;
byC, fco8"*(to= ?iM^ + lf"te=5E^"2S*i" + c
The c6mbination of the results gives
Tsin" cos* xdx = - ^ sin^ x cos*x
" ^ sin* x cos^ x - ^ sin x cos^ x
+yJj
sinx cos^x + ^ sinx cosx + ^x + c.
The formulsB might have been applied in other orders, for example CACAA,
CCAAA, etc.
Ex. 8. Solve Exs. 2, 3, 4, 5 by means of the reduction formulse.
* Another method of deriving these formulse, namely, by integrating by
parts, is given in Note C, Appendix. Other formulse of reduction can be
obtained by connecting
I sin* X cos" X dx with I sin'"-2x cos""+2 x dx, isin""+2 x cos"-* x dx
in the manner described above. The formulse for these cases may be derived
as an exercise. (See Edwards, Integral Calculus, Art. 83.)
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52.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 113
52. Itan*^ no sec** x dx, |cot^ x cosec*^ x dx.
(a)Reduction to the form jsin^ajcos^ojda?. This may be done
by the substitutions
sin a? 1i
cos a; 1tajia;=
,seco; =
, cot x =-:
"
,cosec a; = "
cos a; coso? smo; smo;
and the integration can then be performed by one of the methods
of the last two articles.
(b)Reduction to an algebraic form.
If tan a; = 2?, I tan^a? sec^'xdx = j2;"*(1z^^ dz.
This is almost immediately integrable if n is a positive even
integer. In this case, I tan*" a? sec" a; (fa?can be reduced to inte-rals
of the form I tan''a:d(tAna?).SeeEx. 1, below.)
tan*" X sec" xdx= 1 2!"~^ (2^ 1)^ dz.
This is almost immediately integrable if m is a positive odd
integer. In this case, I tan*"a; sec" a?da? can be reduced to integrals
of the form |sec* a;c?(seca?).(SeeEx. 1, below.)
The form I cot'*ajcosec"aj dajmay be treated in a similar manner.
Ex. 1. Ttan^xec*xdx=^
Ttan* sec^x
seci^xdx
= (taiTi^x(tm^x+l)d(i9,nx)
= Jtan"x + Jtan*a; c.
Or, Itaii^ X sec* x dx = I tan^ x sec^ x sec x tan x dx
= i (sec2"
1)sec* x d (secx)
= i sec^ X " J 8ec*x + c.
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114 INTEGRAL CALCULUS [Cii.VII. 62-
Ex,3. fsec"g(to, g^ g fcot6xco8ec"x(te-J lan8 X J
Ex.8, ftan^xsec^xda;. Ex. 6. ftaiiTx8ec"xdx.
Ex.4, i cot^ X cosec* X dx. Ex.7. I tan* x sec* x (to.
Ex. 8. Solve some of these examples by using algebraic transformations.
53. Use of multiple angles. When m and n are positive and
one of them is odd, the firstmethod of integration shown in Art.
50 can be employed in the ease of I sin* x cos" a da?. When m and
n are positive and both even, the use of multiple angles will
aid the process of integration. The trigonometric substitutions
that can be employed for this purpose are :
sin X cos a; = ^ sin 2 a;,
cos* a? = ^(1+ cos 2a?),
sin*a;= ^(1 " cos 2aj).
Ex. 1.fsin2
X cos^ x dx =
JJsin^2 x dx
= JJCl co84x)dx
= Jx " ^8in4x + c. (Compare Ex. 1, Art. 61.)
Ex. 2. fsin*x dx. (SeeEx. 6 (a),Art. 46.)
Ex. 8. fcos* X dx. (See Ex. 7 (a),Art 46.)
Ex. 4. fsinsdx. (SeeEx. 6 (6),Art. 46.)
Ex. 6. fcos"xdx. (SeeEx. 7(6),Art 46.)
Ex.6. fsin*xcos2xdx.(See Ex. 2, Ar*. 61.)
Ex. 7. Tsin* cos* X dx.
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65.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 115
54f ^
.
On denoting the integral by /,and
dividing thenumerator and
denominator by cos^ x,
j__r sec'xdx
_
r d{t\""
J a^ 4- 6Han2 a;
~
J a^ 4- "
:an x)Wt^,u^x
On substituting u for tana;, integrating, and replacing u by
tan X, there is obtained,
ab \ a J
eer dx C doc
J a + 6 COS ac' J a + 6 sin a;*
Since cos x = cos^ ^" sin*^,nd cos^ ^+ sin^^= 1"
c ^=r
'.
^ m
On dividing numerator and denominator in the second member
by
cos*^nd reducing,
sec* ?dx
r djx^
1 r 2
Ja + "cosaja^bJ ^^^
.^ + ^
2 a-6
(fAan^^
^-"^"^tan^^ + ^Li:^2 a " 6
"1 oy
^^ 11"
T5" 3.ccord-
ing as a is greater than b, or less than b. Hence,
ifa"", f" ^=
_l_tan-Y\^tan2Va + 6 cos a;. Va* " 6* V a + ^
V
INTEGRAL CALC. " 9
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116 INTEGRAL CALCULUS [Ch.Vn. 55-
ifa " 6, I "
-^
= "
,
log "^""-
=
_l_tanh-fJ|H"tan?YOn introducing the half-angle as before, and dividing numerator
and denominator by cos* % as in the case justconsidered,it
will
be found that,
J a-{-b sinx J
dx
a/'cos^lsin^l'j2 6 sin|cos|
=/:sec^^da;
a-f-
2 6 tan I-ha tan*?
a J I
This is in the form 1"-?".,or f^
^
^ according as a is greater
J z^-^-crJ z^ " cr
or less than b. Hence,
if a " 6, r"
"^.
" =^
tan-^
J a -h 6 sma;
-^^2_
52
a tan - + 5
a tan -
-f6 " V o'* " a^
if a " 6, I "
-^" :=
,
log
Ja-hbsmx V6*-a* % tan ?+ 6 + V6^^^
V6*-a*: coth-i
atan^+6
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56.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 117
In working the examples it is preferable to follow the method
employed above, and not to use the results that have justbeen .
found as formulae for substitution.
Ex. 1 f "^*-
v^ A C dx
,.1.f ^^ . Ex.4, f"J3-2s!nx J6- 3cosa;
^"2. f^ f^. " Ex.6, f" -
J2 + 38in2x J 4 + 5 cos 2 X
8. (- p Ex.6, f"J5 + 3COSX J 4-+ 5 sin 2 "
Ex. 7. When a = 6 in this article,find the integrals.
56. \")^"innxdXf (e^eosfixdx.
On integrating e'^ sin nxdx by parts, first taking e^dx for dv,
and then taking amnxdx for dv, thereare
obtained
Te"smnxdac =
^'^^^^- - fe osnxdx, (1)
Ce^smri^dx-^^-^^^^-h-C^cosnxdx,
(2)
The integral in the second members of (1),(2)can be elimi-ated
by multiplying the members of (1)by - and the members
n
^
of (2)by -, and adding the results. When this is done it will
be found that
/'"-sinr^d^ = '^^^(^^^^^^^^21^. (3)
Similarly, on integrating e"*" cos nxdx by parts, first taking
e'^dx for dv, and then taking cos na; da; for dv, and eliminating
I e*** sin nx dx which has thus been introduced, there will be
obtained
J a* + ""^ ^
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118 INTEGRAL CALCULUS [Ch. VII.
The result (4)can be obtaiued by eliminating ief^^mnxdx
from
(1)and (2).It can be deduced
also
by
substitutingthe
result (3)in (1)or in (2). It is, however, preferable to deduce
it directly by integrating by parts.
As in Art. bb the student is advised to work the examples
by the method followed above, and not to use (3)and (4)as
formulae for substitution.
X. 1. fe'sinxdx. Ex. 4. J-x. 1. Je'sinxdx. Ex.4.
\^2^dx.
Ex. 2. Je'cosxdx. Ex. 6. f^"to.
Ex. 3. (e^cm^xdx. Ex. ". ie'Go^^xdx,
57. Tsinrnxos n^vcf a?, rco8fna?co8na?c7;v,Jsinm^vsiima^c
Since, by trigonometry,
sin mx cos nx = ^ sin (m-{-n)x+ \ sin (m "
n)x,
/,cos (m
-{-n)xcos (m "
n)xsin mx cos nxdx = "
"
-)!"
-^^ -^4"
2(m +
n)2(m "
n)In a similar way it can be shown that,
/,sin (m
-{-n)x sin (m "
n)xcos mx cos nx dx =
"-i" "
-^ -\"
-i^,
2(m + 71)^
2(m-w)
/,
sin (m 4- n) x sin (m "
n)xsm ma? sin nx dx =
^r^^" " " f"
-\^ ^
2(m4-w)
^
2(m-n)
Ex. 1. jcos 3 a; sin 6 X dx. Ex. 4. icos 3 x cos Ja;dx.
Ex. 2. rcos4xcos7xdx. Ex. 6. fcos |x sin Jx dx.
Ex.3. (sin5xsin6xdx. Ex.6, fsiny'^
x sin^s^y
dx.
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CHAPTER VIII
SUCCESSIVE INTEGRATION. MULTIPLE INTEGRALS
58. Successive integration. It has been seen in the differential
calculus that successive differentiationith respect to x is some-times
required in the case of functions of the form
%=/(a?);
and that successive differentiation with respect to both x and y
may be required in the case of functions of the form
u=f(x,y).
On the other hand, the reverse process called successive inte-ration
is sometimes necessary. This chapter will be concerned
with describing the notation that is used in "multiple integrar
tion,"as it is
often termed ; andit
will show,by
examples,how
successive integration is introduced and conducted. Arts. 61,
62, 63, contain applications of multiple integration to the measure-ment
of areas in rectangular coordinates, and of volumes in rec-tangula
and polar coordinates. Plane areas in polar coordinates
and ciirvilinear surfaces will be found by means of multiple
integration in Arts. 67, 75.
59. Successive integration with respect to a single independent
variable.
Suppose that fii^)= |/(^)^^" (1)
f2(x)=fA(x)d^, (2)
119
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120 INTEGRAL CALCULUS [Ch. Vlll.
and M^)=fMx)dx. (3)
Since /j(x)
=f[fi")]x,
itfollows from (1)that f^(x)=C [Cf(x)dx]x ; (4)
and since f^(x) j\f^(x)\dx,
it follows from (4)that /g(a;) fjf[Cf(x)dx]x ]-x, (5)
The second member of (4)is usually written in a contracted
form, namely,
jjf{x)dxdx,r
^jf{x)^, (6)
in which da? means {dx)^,nd not d (a?).
Similarly, the second member of (5)is usually written
CCCf(x)dxda:dx,r CfCf(x)da?, (7)
Integral (6)is called a double integral, and integral (7)is called
a triple integral. In general, if an integral is evaluated by means
of two or more successive integrations, it is called a multiple in-egral.
If limits are assigned for each successive integration, the
integral is definite ; if limits are not assigned, it is indefinite.
Ex. 1. Determine the curve for every point of which the second differ-ntial
coeflBcient of the ordinate with respect to the abscissa is 8.
The given condition is expressed by the equation
(1)^
= 8.
"mdx
whence, d(^] =
\dorJ
This may be written-^
" = 8 ;
Sdx.
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59.] SUCCESSIVE INTEGRATION 121
Integrating, (2) ^ = 8 x + c,
dx
whence, dy = (Sx -^ c)dx.
Integrating again, (3) y = 4 x^ + ex + k.
This is the equation of any parabola that has its axis parallel to the y-axis
and, drawn upwards, and itslatus-rectum equal to 4. All such parabolas will
be obtained by giving all possible values to c and Jfc,he arbitrary constants of
integration. Two further conditions will serve to make c and k definite.
For instance, suppose that the tangent to the parabola at the point whose
abscissa is 2, isparallel to the x-axis ; and also that the parabola passes through
the point (3,6). By the former condition,
^= 0whenx = 2;
dx
and hence, by (2), 0 = 8 . 2 + c,
that is, c = " 16.
Equation (3)then becomes y = 4 x^ " 16 x + A.
Also, since the parabola passes through the point (3,5),
6 = 4. 32-16. 3 + A;;
whence. A;= 17.
Therefore the equation of the particular parabola that satisfiesthe three
conditions above is
y = 4x2-16x + 17.
The given relation (1)might have been written in the differential form,
d^ = Sdx^,
and y expressed in the form of a multiple integral, namely,
whence on integrating, = ( (Sx + c)dx
= 4x2 + cx + A;.
The former solution is better because it shows all the steps more clearly.
Ex. 2. If 8 represents distance measured along a straight line, and t time,
" is the velocity of a body that moves in the straight line, and " is its
dt dfi
acceleration or rate of change of velocity. In the case of a body falling in
a vacuum in the neighborhood of the earth^s surface, the acceleration or rate
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122 INTEGRAL CALCULUS [Cii.VIII.
of increase in the velocity is constant and equal to about 32.2 feet-per-second
per second. The number 32.2 in this connection is denoted by the symbol
g. Let it be required to determine 8 from the known relation,
'(I)his may be written " =- " = g.
Using the differential form, "i(-)= fl'*"
da
and integrating, (2) " = gt + c,
at
in which c is an arbitrary constant of integration.
Writing the latter equation in the differential form,
d8=("gt^c)dt,
and integrating, (3) " = ^ gt^ + c" + Jfc,
in which k is another arbitrary constant of integration. In order that the
constants c, k may have definite values, two further conditions are required.
For instance :^
(a) Suppose that the body fallsfrom rest, and that the distance ismeasured
from thestarting point.
In this case, s = 0, and " = 0, when * = 0.
Hence, substituting in (2), 0 = 0 + c,
that is, c = 0 ;
and, substituting in (3), 0 = 0 + 0 + ik,
that is, ik= 0.
Therefore, the distance through which a body fallsin a vacuum on starting
from rest is \ gt^^ in which g is about 32.2 and t is the duration of fall in
seconds.
(6) Suppose that the body has an initial velocity of 8 feet per second, and
that the distance is measured from a point 12 feet above the starting point.
By the last condition, " = 12 when t = 0;
and hence, by (3), 12 = 0 + 0 + jfc,
whence k = 12.
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69-60.] SUCCESSIVE INTEGRATION 123
By the other condition, " = 8 when " = 0 ;" dt
and hence, by (2), 8 = 0 + c, that is, c = 8.
Therefore, under these conditions,
8 = \gV^-\.^tJr12.
The known relation (1)might have been written in the differential form,
d^s = gdt^ ;
from this, * = i\9dt^
whence, on integration, " = \{gt-\-c)dt
= hgt^+ ct + k.
Ex. 3. Evaluate C C C^ (dxy.
The integrations are made in order from right to left. Thus, if / denote
the integral,
'=n"[?j:(-'-rf"*"'
= 16.
Ex. 4. Evaluate CCCoi*(dxy. Ex. 6. Evaluate VCC^i^^)^-
(Compare Exs. 4, 6, with Ex. 3.)
d^vEx. 6. Determine all the curves for which " ^ = 0.
dx^
Ex. 7. Find the curve at each of whose points the second derivative of
the ordinate with respect to the abscissa is four times the abscissa, and
which passes through the origin and the point (2,4) ..
Ex. 8. Find Prf'^^C^^O^-x. 9. O f^psin d^)8.
GO. Successive integration with respect to two or more indepen-ent
variables. In this article the notation commonly used in this
kind of integration will be described ; and, in preparation for the
next article, a few examples will be given so that the student
may become familiar with the notation.
Suppose that fi(x,, z)= \f(x, y, z)
dz, (1)
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124 INTEGRAL CALCULUS [Ch. VIII.
the integration indicated in the second member being performed
as if X, y were constants. (Itwill be remembered that if Xy y,---,
are independent variables, differentiation of F(x, y, "..,)ith
respect to one of the variables, say a?, is performed as if the others
were constants.)hen, suppose that
/2(",, z)= J/i(",, 2;)
y, (2)
the integration now being performed as if a;^ 2; were constants.
Again, suppose that
M^y Vi 2)= J/2("," z)
dx, (3)
the integration being performed as if y, z were constants. Equa-ion
(2),y virtue of equation (1),an be put in the form
/2(",y 2)= JlJV(^'^ ^) A
^y 5 W
and equation (3),y virtue of (4),an be wi'itten,
/aCaJ,, z)=J IJ IJ/(^'^ ^) 2=^2I^. (5)
The bracketing in the second member of (5)indicates that the
differential coefficient,f(x,y, z),is integrated with respect to z ;
that the result of this integration is then integrated with respect
to y ; and that, finally,the result of the last integration is inte-rated
with respect to a?. In the notation usually adopted, the
second member of (5)is abbreviated by removing the brackets,
and the order of the variables with respect to which the integra-ions
are made, is indicated by the order of the respective differ-ntials
of the variables beginning at the right and going toward the
left.Thus, the abbreviated form of the second member of (5)is
ffffi^yy z)dx dy dz. (6)
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60.] SUCCESSIVE INTEGRATION 125
This is a triple integral. Similarly, the double integral in the
second member of (4)is generally written,
Jff(^yyyZ)dydz.As to the integration signs, the first on the right is taken with
the first differential on the right, which is dz in (6)above, the
second sign from the right is taken with the second differential
from the right, the third sign from the right is taken with the
third differential from the right, and so on. It is well to note
this usage, because attention must be paid to it when limits of
integration are assigned to a;, y, z* In some of the examples
below, and often in practical problems, the limits for one variable
are functions of one or more of the other variables.
Ex.1. Evaluate
) )
T xy^dxdydz.
If /denote the integral,
J= rf [zYxy^dxdy ^( Cxy^dxdy
Ex.2. Evaluate ] \' xy^dxdy.
* The notation described above is not universally adopted, but it is the
one most frequently used.
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126 INTEGRAL CALCULUS [Cn. VIII.
61. Application of successive integration to the measurement of
areas : rectangular coordinates. In this and the two following
articles, proV)lems are solved which show applications of succes-sive
integration. In some of the examples there may not be
any special advantage in resorting to double integration, for the
reason that a single integration may suffice. They are, however,
given to the student for the purpose of making him familiar with
an instrument for solution which may sometimes be the only one
possible. It will be found that the elements in the summations
which follow are infinitesimals of a higher order than those which
have been met with heretofore.
Fig. 81.
Ex. 1. Find the area included between the parabolas whose equations are
3?/2 = 25", and 5x2 = 9y.
The parabolas are VOP^ WOP. Their points of intersection, 0, P, are
(0,0), (3,5). Taking any point Q within the area as a vertex, construct
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61.] SUCCESSIVE INTEGRATION 127
a rectangle whose sides are parallel to the axes of coordinates and are equal
to Ax, Ay..
Produce the sides which are parallel to the x-axis until they
meet the curves in X, ilf,G^ B, thus forming the strip LGBM^ and produce
ML to meet the y-axis in B. On LM construct the rectangle HM^ giving it
a width Ay.
x==RM
Area of the rectangle HM"
limit^^^^ ^ Ax Ay.
Both y and Ay remain unchanged throughout this summation. Now,
BL=^,and i?i"f=
3\|.
Hence, area HM= \ \^
dx\Ay (1)
'=-"
(W|-S)a. (^)
As Ay approaches zero, the rectangle HM approaches coincidence with
the infinitesimal strip GM, and, in the limit, HM coincides with GM, Also,
the area OLPMO is the limit of the sum of all the strips similar to LGBM
lying between 0 and P, when Ay is made to approach zero as a limit.
Therefore,
y=py_
area OLPMO =
limit^y^o/^(3^'|^)^2/
= 5.
If the linear unit is.an inch, the answer is in square inches. On substi-uting,
for lsJ^--^^\m3) its value as shown by (1) and (2),there is
obtained,
area OLPMO = ^'J_',[^^']^y
=
"f^''dydx. (4)3y_
25
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128 INTEGRAL CALCULUS [Ch. VIII.
The latter is the customary abbreviated form which indicates that the first
integration is made with respect to x between the limits
_iLand 3-\/?or
", and that the result obtained thereby is to be integrated with respect to y
between the limits 0 and 5. The element of area in (4),namely dydx^ is an
infinitesimal of the second order.
Another way of performing the double summation required in adding up
all of the elements of area like dy dx, may be described as follows. Sum all
of these elements that are in the vertical strip ST, and then sum all of the
vertical strips in OLPMO. In the firstsummation, x and dx do not change,
and the upper and lower limits of y are 5-v/-,^
respectively ; in the second
summation, the limits are the values of x at 0 and P, namely 0 and 3. This
double summation is indicated by the double integral
s:s:fxdy.
9
This, on evaluation, gives an area of 6 square units as before.
The area OLPMO might have been expressed in terms of single integrals.
For
OLPMO = OLPN- OMPN
Jo ^3 Jo 9
= 10 - 6 = 5.
Ex. 2. Solve Exs. 7-11, Art. 29, by this method.
62. Application of successive integration to the measurement of
volumes: rectangular coordinates. If the equation of a surface
is given in the form
the volume can usually be determined by means of three successive
integrations. In the particular case of solids of revolution, the
volume can be found by a single integration. This was shown
in Art. 30. In Art. 61 the element of area was dydx, the area
of an infinitesimal rectangle each side of which was an infini-esimal.
In the case now to be considered, the element of volume
will be the volume, dxdydz, of an infinitesimal parallelepiped
each of whose infinitesimal edges is parallel to one of the axes
of coordinates. This will be illustrated in Ex. 1.
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61-62. J SUCCESSIVE INTEGRATION 129
Ex. 1. Find the volume of the ellipsoid whose equation is
a;2 y2 g2
Let 0-ABC be one eighth of the ellipsoid whose volume is required.
Then OA = a, OB = b, OC = c. Take IL an infinitesimal distance dx on
Pio. 82,
OX, and through /, L pass the planes HIJ, KLM perpendicular to OX
Take EF of an infinitesimal length dy on ML^ and complete the infinitesimal
rectangle EFGD. Through the lines DE, GF pass planes parallel to the
plane ZOX which intersect the curvilinear surface HJMK in the infinitesi-al
arcs BV, ST. Through a point D' on DB, DD' having an infinitesimal
length dz, pass a plane parallel to the plane XOY. The infinitesimal paral-elepiped
D'F^ whose volume is dxdydz, will be taken for the element of
volume. The solid 0-AOB is the limit of the sum of parallelepipeds of this
kind. This limit will now be determined.
First, the volume of the vertical rectangular columnBF
willbe found
by
adding together all the infinitesimal parallelepipeds such as D'F which are
included between DEFG and BSTV.
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130 INTEGRAL CALCULUS [Ch. VII!.
Second, the volume of the slice HIJMLK will be found by adding together
all the infinitesunal rectangular columns like BF which are erected between
IL and JM.
Third, the volume of O-ABC will be found by adding together all the
infinitesimal slices like HIJMLK that lie between OCB and A,
In the addition of the infinitesimal parallelopipeds from DEFG to BST T,
z alone varies, and it varies from zero to DB.
Vol. BF = dydz. 0)
In the addition of the vertical columns from IL to MJ, y alone varies,
and it varies from zero to IJ,
,;Yo\,HIJMLK
=\f f(fe
dy\dx,lyioL. 0 J J
(2)
In the addition of the slices between OCB and A, x varies from zero
to OA.
x = OA , y = IjrM=DR~= UA I y " lJ H=DR ^
". Vol. O-ABC = f If ^dzdyidx.
Writing this in the usual manner.
x=OA y = TJz = DR
Vol.
0-ABC= f r J dxdydz.
(3)
(4)x = 0 y = 0 " = 0
If the coordinates of B are a:, y, z, it follows from the equation of the
surface that
DB = z,
At the point J, z = 0; and hence.
'-^"-g-J
Also, OA = a
Therefore (4)becomes
Vol. O-^JJOJo Jo Jo
(?xdy dz.
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02-63.] SUCCESSIVE INTEGRATION 181
On making the integrations in their proper order, it is found that
Vol. 0-ABC = iirabc.
Hence, the volume of the whole ellipsoid = fnabc.
Note 1. The volume of an infinitesimal parallelopiped is an infinitesimal
of the third order, the volume of a vertical column is an infinitesimal of the
second order, and the volume of a slice is an infinitesimal of the firstorder.
NoTB 2. Equally well, the planes bounding an infinitesimal slice might
have been taken perpendicular to either OZ or 0 Y,
Note 3. On putting a = b = c, the volume of a sphere of radius a is
found to be I ira*.
Ex. 2. Find the volume of the ellipsoid given in Ex. 1 : (a) by taking
the infinitesimal slice at right angles to OT; (") by taking it at right angles
to OZ.
Ex. 8. Determine the volume of a sphere of radius a by the method of
this article.
Ex. 4. Find the volume bounded by the hyperbolic paraboloid " =^,
the xy-plane and the planes z = a, x = A, y = b, y = B.^
Ex. 6. Find the volume of the wedge cut from the cylinder x^-\-y^ = a^
by the plane z = 0, and the part of the plane z = x tan a for which z is
positive.
Ex. 6. Find the entire volume bounded by the surface
\a \b \c
Ex. 7. The center of a sphere of radius a is on the surface of a right
blinder the radius of whose base ii
cylinder intercepted by the sphere.
cylinder the radius of whose base is -. Find the volume of the part of the
63. Farther application of successive integration to the measure-ent
of volumes: polar coordinates. The illustration in this
article is given, because the use of polar coordinates in dealing
with solids is often advantageous. It will be necessary to employ
these coordinates in solving some of the problems in Arts. 77, 79.
Ex. 1. To find the volume of a sphere of radius a by means of polar
coordinates. Let a point O on the surface of the sphere be the pole, the tan-
INTEGRAL CALC. " 10
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63.] SUCCESSIVE INTEGRATION 183
I of sphere z= ft-I
ftolume ( : sphere = f' f f "''"^^M^ "am0d0
tf=0 t "^=rOL.
r=
0J
J
= ?|?p r^cos"^ sin ^ (W (i0
3 Jo
Ex. 2. Find the volume of sphere of radius a by thi^ method, on letting
the XO Y plane pass through the center.
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CHAPTER IX
FURTHER GEOMETRICAL APPLICATIONS. MEAN
VALUES
64. The calculus has already been employed for the derivation
of the equations of curves in Art. 32, for the determination of
the areas of curves in Art. 27, for the determination of volumes
of solids of revolution in Art. 30, and for the determination of
volumes of solids in a more general case, in Arts. 62, 63. Carte-ian
coordinates were used in all but the last of these applica-ions.
This chapter will consider the derivation of the equations
of curves and the measurement of areas in cases in which polar
coordinates are employed. Special cases in areas and volumes
are taken up in Arts. 68-70. The measurement of the lengths of
curves for both Cartesian and polar coordinates is considered in
Arts. 71, 72 ; and the measurement of surfaces is discussed in
Arts. 74, 75. The subjectf mean values is treated in the lasttwo articles of the chapter.
65. Derivation of the equations of curves in polar coordinates.
Let the equation of a curve in polar coordinates be
/(r,^)=0.
It is shown in the differential calculus that, if (r,$)be any point
on the curve,\f/
he angle between the radius vector and the tan-ent
at (r,0),and "^the angle that this tangent makes with the
dBinitialline,then tan \l/ r "
,
dr
134
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64-66,] FURTHER GEOMETRICAL APPLICATIONS 186
the length of the polar subtangent
the length of the polar subnormal
__dr~de
Ex. 1. Find the curve in which the polar suhnormal is proportional to
(is times)the sine of the vectorial angle.
drIn this case, " = k sin $.
de
Using the differential form, dr = k sin $d$j
and integrating, r = c " k cos ^.
If c = If, r = K (1 " cos ^),
the equation of the cardioid.
Ex. 2. Find the curve in which the polar subtangent is proportional to
(isK times)the length of the radius vector.
Ex. 8. Find the curve in which the angle between the radius vector and
the tangent is n times the vectorial angle. What is the curve when n = 1 ?
when n = J ?
66. Areas of curves when polar coordinates are used: by single
integration. Let AB be an arc of the curve r=f($),and suppose
that angle ^OX=:a, angle BOL = fi. The area of AOB is re-quired.
Divide the angle AOB into n parts, each equal to A^; then
The sector AOB will thus be divided into n sectors, which have
equal angles at 0. Let POQ be one of these sectors. About 0
as a center, and with a radius equal to OP^ describe through P a
circular arc PPj, which intersects OQ in Pi ; and about the same
center 0 with OQ as a radius describe an arc QQi,which meets
OP in Qi. The area of the sector POQ is greater than the area
of the "interior'' sector POPi, and is less than the area of the
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136 INTEGRAL CALCULUS [Ch. IX.
"
exterior"
one QOQi. About 0 as a center, and through each of
the points in which the arc AB is intersected by the lines that
divide the angle AOB into equal parts, let circular arcs be drawn
which intersect the adjacentlines of division on each side, as
Fio. 84.
shown in Fig. 34. There is thus obtained a set of exterior circu-ar
sectors like QiOQ, and a set of interior circular sectors like
POPy. It has been seen that each sector POQ of the figure AOB
is gi*eater than the corresponding interior sector POPi, and less
than the corresponding exterior sector QOQi ; that is,
sector POPi " sector POQ " sector QiOQ.
Therefore the sum of the sectors POQ, which is AOB, is
greater than the sum of the interior sectors, and less than the
sum of the exterior sectors ; that is,
^POPiAOB
"^QOQ,.
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66.] FURTHER GEOMETRICAL APPLICATIONS 137
In the limiting case, when the number of sectors POQ becomes
infinite,that is when A$ approaches zero, the sum of the areas of
the interior sectors, and the sum of the areas of the exterior sectors
approach equality. For, the difference between these two sums is
equal to the area of AMM'A\ which is ^(OM^ - 0A^)^$, and
can therefore be made as small as one pleases by decreasing A^.
The area AOB always lies between these sums ; and hence,
area AOB =
limit^^^oy^^OPi.If the coordinates of P be denoted by r, $, the area of
POPi=ir'Ae.
Hence, are'a AOB = limit^^
=0/
i ^^^ 5 ".
that is, area AOB =i fVde,
by the definition of a definite integral. The element of area for
polar coordinates is thus ^ r^dO,
Ex. 1. Find the area of the sector of the logarithmic spiral whose equation
is r = c"^, between the radii vectores for which ^ = o, tf= /3.
Area POQ = U^f^dB
4a
ri, r2 being the bounding radii
vectores.F16. 86w
Ex. 2. Find the area of one loop of the lemniscate r^ = a^C082 ^.
The area of one half the loop,^
'
OMA =
i(f^de,
between proper limits for ^, which must be determined.
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188 INTEGRAL CALCULUS [Ch. IX.
The initial and final positions of the radius vector are OA and OL the tan-ent
to the arc OM at 0. For r = 0, the equation of the curve giv^es
0 = aaco82^;
and hence,
Fig. 86.
The positive sign indicates the position of OL, and the negative sign that
of ON.
Hence, area OMA
-^S!^de
Hence,
= "f^cos2^d^
2 Jo
4*
area of loop OMARO = " .
2
Ex. 8. Find the area of a sector of the spiral of Archimedes, r = a0,
between $ = a, 0 = p,
Ex. 4. Find the area of the part of the parabola r=^a sec^ - intercepted
between the curve and the latus rectum.2
Ex. 6. Find the area of the cardioid r^ = a^ cos -"
2
Ex. 6. Find the area of the loop of the folium of Descartes,
ajs + ys _ 3 fjrg^y. 0.
(Hint: Change to polar co5rdinates, thus obtaining r =
3 g sin ^ cos g
cos* e + sin* e; and
then change the variable B by putting z = tan $.)
Ex. 7. Show that the area bounded by any two radii vectores of the
hyperbolic spiral 7*8 = a, is proportional to the difference between the lengths
of these radii.
Ex. 8. Show that the area of a loop of the curve r^ = a^ cos n$ is " .
n
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66-67.] FURTHER GEOMETRICAL APPLICATIONS 139
67. Areas of curves when polar coordinatesare used: by double
integration. The areas of curves whose equations are given in
polar coordinatescan be found by double integration,n a man-er
analogous to that used in Art. 61. Example 1 below will
serve to make the method plain. Successiveintegrationin two
variables,polar coordinates,willalsobe required in Arts. 77, 79.
I
Ex. 1. Find by doable integrationthe area of the circlewhose equation
is r = 2acos^.
Let OLAN be the given circle,0
being the pole and OA the diameter
2 a. Within the circle take any
point P with coordinates (r,0),Draw OP and produce ita distance
Ar to S, Revolve the lineOPS about
0 through an angleAS to the position
OQR^ Then
area PQBS= i[(""^*')*^} ^^
z=:rArAe'^i(^AryA9.Fig. 87.
Produce OP, OQ to meet the circlein Jf, G. The area of the sector MOG
will be found by adding allthe elements PQBS therein,and the area of the
semicircle OLMA will be calculated by adding all the sectors like MOG
that itcontains.
Area.MOG = 2iPQRS
r=ajr
=
\imitj:^r=o/](rArAeJ(Ar)"^)r = 0
r=OJf
= limit^= 0 / rArAB
r=ii
by a fundamental theorem in the calculus,*
r=OM
and hence, area MOG = irdrAd,
* See Note D, Appendix.
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140 INTEGRAL CALCULUS [Ch.IX.
in which the integrationisperformed with respect to r. Hence,
B=TOX r=OM
area OLMA = f Tfr dr IcW"=0 r=0
Jo Jo
2
Hence, the area of the circleiswa^.
The area of OMA can alsobe obtained by findingthe area of the circular
stripLG whose arcs are distantr, r + dr from 0, and then adding allof the
similarconcentriccircularstripsfrom O to A. The angle LOG = cos"^ ^" It
willbe found that
JO
"^---2
2a
area OMAJo Jo
'
rdrd0 = ^ as before.
Ex. 2. Find by double integrationthe area of the circleof radius o, the
pole being at the center ; (1)by adding equiangularsectors ; (2)by adding
concentric circularstrips.
I
68. Areas in Cartesian coordinateswith oblique axes. In this
case the method of findingthe area issimilarto that in Art. 61.
Let the axes be inclinedat
an angle cu, and construct a
parallelogram whose sides
are parallelto the axes and
have the lengthsAa5,
Ay.The area of this parallelo-ram
is
Ax Ay sino).
The whole area is the limit
of the sum of all parallelo-rams
that are constructed within the perimeter of the figure
when
Ax
and
Ay approach zero. Hence, the area is the value of
the double integral
IIsin0) c2a;y, that is,sina" | idx dy,
Fie. 88.
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142 INTEGRAL CALCULUS [Ch. IX.
When 35 = 0, ^ = 0; and when " = 2ira, ^ = 2ir.
Also, (to = o (1 " cos 0)dd.
If these values fory, dx, and the limits, be substituted in the integral
above, it becomes
area =
a2f''(I
cos^)"d^
=
a2j^''(l-2cos^i"f^)(W
That is,the area is three times that of the generating circle.
Ex. 8. Find the area of the ellipse -3 +p
= 1- (Compare Ex. 3, Art. 27.)
(Hint: put x = acos0, then y = 6 sin0.)
Ex. 4. Find the volume of the solid generated by the revolution of a
complete arch of the cycloid of Ex. 2 about the x-axis.
70. Measurement of the volumes of solids by means of infinitely
thin cross-sections. In Art. 30 the volume of a solid of revolution
was determined by finding the volume of an infinitely thin slice
of the solid,the slice being taken at right angles to the axis of the
figure, and the sum of the volumes of all such slices being then
found. This method can be extended to other figures besides
figures of revolution. Some convenient line is chosen, and an
infinitesimally thin slice of the solid is taken at right angles to
this line. If the area of a face of the thin slice can be expressed
in terms of its distance from some point on the line, the volume
of the slice can be expressed in terms of this distance ; and from
this, the sum of the volumes of all the slices can be found. For
example, let the chosen line be taken for the axis of a?, and sup-ose
that the area of a face of a thin slice at right angles to this
line is /("). Let the thickness of the slice be Aa?. The volume
is (asin Art. 30) the limit of the sum of an infinite number of
infinitesimal cylinders whose volumes are of the form /(a?)a5.
That is,
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60-70.] FURTHER GEOMETRICAL APPLICATIONS 143
voluiDe of the solid = limits^o
S f(x)
=Jf(x)dx,in which the limits of integration are determined from the figure.
Ex. 1. Determine by this method the volume of the ellipsoid
(The studentis
advisedto
makea figure.
)At a distance x from the
center
cut out a very thin slice at right angles to the x-axis, and let its thickness be
dx. The face of this cylindrical slice will be an ellipse whose semiaxes are
These values are deduced from the equation of the ellipsoid.
The area
ofthis
ellipse=
irftc/l -
^V
Hence, the volume of the slice =irbc[1 " ^ j
dx ;
and therefore, volume of ellipsoid = irbci *(l")d!K
= I wobc.
Ex. 2. Find the volume of a sphere of radius a by this method.
Ex. 8. Find the volume of the torus generated by revolving about the
o^axis the circle x^ + (y " b)^= a^, in which 6 " a.
Ex. 4. Find the volume of a pyramid or a done having a base B and a
height h.
Ex. 5. Find the volume of a right conoid with a circular base and alti-ude
h, the radius of the base being a.
Ex. 6. A rectangle moves from a fixed point, one side varying as the
distance from this point, and the other as the square of this distance. At
the distance of 2 feet, the rectangle becomes a square of 3 feet. What is
the volume then generated ?
Ex. 7. Given a right cylinder of altitude h, and radius of base a.
Through a diameter of the upper base two planes are i"assed touching the
lower base on opposite sides. Find the volume included between the planes.
Ex. 8. Find the volume of the elliptic paraboloid 2 x =^
+^
cut off by
the plane x = h, P Q
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144 INTEGRAL CALCULUS [Ch. IX.
71. Lengths of curves: rectangular coordinates. To find the
length of a curve is equivalent to finding the straight line that
has the same length as the curve. For this reason the measure-ment
of its length is usually called **the rectification of the
curve."* The deduction here made of the integration formulae
for the length of a curve depends
upon the definition that integra-ion
is a process of summation.
The equation of a given curve
is /(",y)= 0 ; it is required to
find the length a of an arc AB,
A being the point (xj,yi),be-ng
the point (ajj,2)-On the
curve take any two points P, Q
whose coordinates are ", y, and
X-f
Aoj, y -f^y. Draw the chord PQ and make the construction
indicated in the figure.
The chord PQ = V(Aaj)*-f (Ay)* (1)
(2)
As Q approaches infinitely near to P, that is, when Aa? ap-roache
zero, the chord PQ approaches coincidence with the
arc PQ. It is shown in the differential calculus that if Aa? is
an infinitesimal of the first order, the difference between the
* In 1659 Wallis (see footnote, Art. 27) published a tract in which he
showed a method by which curves could be rectified, and in 1660 one of his
pupils, William Neil, found the length of an arc of the semi-cubical parabola
01?= aiP.This is the firstcurve that was rectified. Before this it had been
generally supposed that no curve could be measured by a mathematical proc-ss.
The second curve whose length was found is the cycloid. Its rectifi-ation
was
effectedby Sir Christk)pher Wren
(1632-1723)and publishedin
1673. This was before the development of the calculus by Leibniz and
Newton.
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71.] FURTHER GEOMETRICAL APPLICATIONS 145
arc and its chord is an infinitesimal of at least the third order;
that is,
arc PQ=
chord PQ-^h, (3)
in which i^is an infinitesimal of the third order when Ao? is an
infinitesimal of the firstorder.
Therefore,s=S(arcsPQ)
.lMit^.,-"yh^'^by a fundamental theorem.* As Ax approaches zero, "
^ in gen-Ax
eral approaches a definite limiting value, namely, --^.Therefore,
by the definition of a definite integral.
In applying this formula it will be necessary to express
^l+(-^]n terms of x before integration is attempted.
Instead of being put in the form (2),equation (1)may be
given the form,
chord PQ
=yJlf^jAy.
By the same reasoning as above, it can then be shown that
in which -i/l ( ) must be expressed in terms of y before inte-ration
is performed. Formula (4)or formula (6)will be used,
* See Note D, Appendix.
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146 INTEGRAL CALCULUS [Ch. IX.
according as it is more convenient to take x or y for the inde-endent
variable.
If A" denotes the length of the arc PQ, it follows from (3)that
Therefore, by the differentialcalculus.
t-yHM-
whence, ds
=\/l+( 3^ ]^"
Similarly,
ds^yjl-hl^)
y:
In order to recall formulae (4)and (5)immediately whenever
they may happen to be required, the student need only remember
the construction of the triangle PQRy and let its sides become
infinitesimal.
Ex. 1. Find the length
ofthe
circle whose equationis 35^
+ y*= a^.
Let AB be the firstquadiantal arc of the circle.
In this case,S^
= _
?.
dx y
m^ "" ^ = f-y/i(!)"*/.'a/m^*dx
0 Va'^ - x3
2*
Therefore, the perimeter of the circle (=4 AB) = 2 ra.
Ex. 2.Find the
lengthof the arc of the parabola from the vertex to the
point (xu Vi). Find the length of the arc from the vertex to the end of the
latus rectum.
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71-72.] FURTHER GEOMETRICAL APPLICATIONS 147
Ex. 8. Find the length of the arc of the semicubical parabola atf^ x^
from the origin to the point (xi,yi). Also to the point for which x = 5 a.
Ex. 4. Find the lengthof
the arc
ofthe
catenary y=
^("*fe
"}from
the vertex to the point (xi,y{). Also to the point for which x = a.
Ex. 6. Find the length of the arc of the cycloid from the point at which
^ = ^0 to the point at which 9z=e\. Also find the length of a complete arch
of the curve.
Ex. 6. Find the entire length of the hypocycloid x* -
Ex. 7. Show that inthe ellipse
x = asin0, y = 6co8^,
"pbeing the complement of the eccentric angle of the point (x,y),the arc "
measured from the extremity of the minor axis is
+ y* =
a*.
Jo
- c2 sin* 0 c?0,
and that the entire length of the ellipse is
rl ,
4a I Vl-easin2 0"i0,Jo
in which e is the eccentricity.*
72. Lengths of curves: polar coordinates. The equation of a
curve is /(r,$)" 0, and the length s of the are AB is required, A
being the point (ri,i),and B the
point (rj,2)- On the curve take anytwo points P, Q, whose coordinates
are r,. 6, r + Ar, ^ + A^. Draw OP,
OQ, and the chord FQ. About 0 as
a center, and with a radius equal toy X JIp
OP, describe the arc PR which inter-ects
OQ injK,and
draw PRi at right
angles to OQ. Then the angle
POQ = ^0, RQ = Ar,^-""^ )i
---^
and arc o^^"^ \ ^'j x_
^" .,
Fig. 40.
PR = rA^.
"This integral, which is known as **the elliptic integral of the second
kind/* cannot be expressed, in a finiteform, in terms of the ordinary func-ions
of mathematics. See Ex. 8, Art. 83.
INTEGRAL CALC. " 11
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148 INTEGRAL CALCULUS [Ch. IX.
It is shown in the differential calculus that when A^ is an infini-esimal
of the firstorder,
PRi = arc PE " ig, an infinitesimal of the third order ;
QEi = QR 4" t'a, an infinitesimal of the second order;
chord FQ = arc PQ " 13, an infinitesimal of the third order.
In the right-angled triangle PBiQ,
chord PQ= VPi^i*
-f BiQ^.
Hence, when A^ approaches zero.
arc PQ- i's= V(PB-isy-h{BQ-^i2y,
or, arc PQ = V(rAd -
hY + (Ar + hf -f*'" (1)
^J^+f^y^^llh^2i,--^''-^''
A^ + i'^ (2)\ ^VA^y A^^ ^(Ad)"^ (A^)"
^ ^'^ ^
which differs by an infinitesimal of at least the second order from
M^)2
Aft
Therefore,
s = SPQ = limit^,.oy^+(|^YA^,
when A^ approaches zero, -" approaches the definite limiting
dr
value "
.Therefore, by the definition of a definite integral,
"=j;'ARif'"-"""
It will be necessary to expressx/^+f-^)
^ terms of $ before
integration is made.^
^ ".t
Similarly, on removing Ar from the radiciai sign in (1),t can
beshown
that
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72-73.] FUBTHEB GEOMETRICAL APPLICATIONS 149
in which -v/l-r^( )must be expressed in terms of r before inte-ration
is made. 1 ormula (3)or (4)will be used according as it
is more convenient to take d or r for the independent variable.
In order to recall these formulae immediately it is only
necessary for the student to remember the construction of the
figure PBQ, and to suppose that its sides are infinitesimal.
Ex. 1. Find the length of the circumference of the circle whose equa-ion
is
Here ^= 0.
d9
Hence.
-r"/^(SF'**=
"/"= 2 ira.
Ex. 2. Find the length of the circle of which the equation is r = 2 a sin 0.
Ex. 3. Find the entire length of the cardioid, r = a(l"
cos^).
Ex. 4. Find the arc of the spiralof Archimedes, r = aO, between the points
(n, ^i),(r2, 2).
Ex. 6. Find the length of the hyperbolic spiral, r0 = a, from (n, ^i)to
(r2, 2).
Ex. 6. Find the length of the logarithmic spiral, r = e^, from (1,0")to
(ri, 1).
Ex. 7. Find the length of the arc of the cissoid r = 2a tan ^ sin 0 from the
cusp (^ = 0) to^=-.4
Ex. 8. Find the length of the arc of the parabola r = q sec^ - from ^ = 0
_ _
r 2
to 0 = 0i; also, from ^ =
--to^= -.
2 2
..":";;;:"
73. The intrinsic 'equation of a curve. Let PQ be the arc of a
given curve, and let s denote its length. Suppose a point starts
at P and moves along the curve towards Q. At the instant
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150 INTEGRAL CALCULUS [Ch.rx.
of starting the point moves in the direction of the tangent
PTi. In passing over the arc PQ the direction of motion
changes at every instant, until at Q the point is moving in
the direction of the tangent QT^. The total change in direc-ion,
as the point moves from P to Q, is measured by the
angle ^ between the two tangents. It will be found that a rela-ion
exists between the distance s through which the point has
moved, and the angle ^ by which the direction of its motion has
changed. This relation between s and ^ is called the intrinsic
^r,equation of the curve. The form
of this equation depends only on
the nature of the curve, and the
choice of the initialpoint P. On
the other hand, the form of the
equation of a curve in other
systems of coordinates, for ex-ample
the rectangular and polar,
depends upon points and lines
that are independent of the curve.
Hence the term " intrinsic.''
To find the intrinsic equation of a curve given in rectangular
or polar coordinates,
(1)Determine the length of arc 8 measured from some con-venient
starting point up to a variable point on the curve.
(2)Find the angle ^ between the tangents at the initial and
the terminal points.
(3)Eliminate the
rectangularor
polar variablesfrom the
equa-ionsthus found.
Fig. 41.
Ex. 1. Find the intrinsic eqaation of the catenary
If the vertex of the curve be taken as starting point,
(1) " = ^(e- e "). [Ex.4, Art. 71.]2
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73.] FURTHER GEOMETRICAL APPLICATIONS 161
Also, since the tangent at the vertex is parallel with the as-azis,
(2)
tan^=gi(e--e"-).
The elimination of x from (1)and (2)gives the required equation between
s and 0, viz.,
8 = a tan ^.
It is easy to extend this result and show that
8 = a [tan("f" "f"i)tan 4"i]
is the intrinsic equation of the catenary when any point A is chosen for the
initial point. The angle ^i is then the angle between the tangent at the
vertex and the tangent at the point A.
Ex. 2. Find the intrinsic equation of the parabola
r= a sec^-.
2
If the vertex of the
parabola
be taken for the initialpoint,
(1) a =
atan?sec^+alogtan/|?) [Ex. 8, Art. 72.]
Also, since the tangent at the vertex makes an angle of - with the polar
axis,^
where 4"'is the angle that the tangent at the point (r,0) makes with the
polar axis. But
2 0' = ^ + T.
Hence, ^ = 2^.
On substituting this value of ^ in equation (1) the intrinsic equation of
the parabola is found to be
" = a tan 0 sec ^ + a log tanf^
+ - j
EXAMPLES.
1. Find the intrinsic equation of a circle with radius r.
2. Find the intrinsic equation of the cardioid r =
a(l"
cos^),the arc
being measured from the polar origin.
8. Find the intrinsic equation of the cycloid
x =
a(e"
sin^),'
y-
) =
a(^"
8in^),lr =
a(l"
cosd),/
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152 INTEGRAL CALCULUS [Ch.IX.
(1) the origin being the initialpoint,
(2) the vertex being the initialpoint.
4. Find the intrinsic equation of the parabola y* = 4jm5,
(1) the vertex being the initialpoint,
(2) the extremity of the latus rectum being the initialpoint.
5. Find the intrinsic equation of the semicubical parabola 3 ay'^= 2 2B*,
taking the origin for initialpoint.
"E6. Find the intrinsic equation of the curve y^a log sec -, taking the
origin for the initialpoint.
7. Find the intrinsic equation of the logarithmic spiral r = a"^"
8. Find the intrinsic equation of the tractrix
y
taking the point (0,c) as the initialpoint.
9. Find the intrinsic equation of the hypocycloid x' + y' = a', taking
anyone
of
the cusps as initial
point.
74. Areas of surfaces of solids of revolution. Suppose that the
surface is generated by the revolution about the a?-axis of the
arc AB of the curve whose equation is y =/(") ; and let the co5r-
R T L s
Fig. 42.
dinates of the points A, B, be a?,, 3/1,and x^, y2" respectively.
Take (Fig.42) any two points on the curve, say P, Q, whose
coordinates are x, y, and x-fAa?, y-j-Ay. Draw the chord PQ
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73-74.] FURTHER GEOMETRICAL APPLICATIONS 163
Fio. 48.
and the ordinates MP, SQ,
and suppose that LM is an
ordinate which is not less,and
that TN is an ordinate which
is not-greater than any ordi-ate
that can be drawn from
the SiTcPNMQ to the a-axis.
(In Fig. 43, LM coincides
with SQy and TN coincides
with RP.) Through N, M,
draw lines PiQ\,P^Qifparallel
to the ovaxis and equal in
length to the arc PNMQ, On the revolution of the arc AB
about OXj each point in AB describes a circle with its ordinate
as radius.
The surface generated by arc PNMQ
^2'jrLMX2ilCQPQy.
and ^ 2irTN x arc PQ.
When Ax is an infinitesimal of the firstorder,
arc
PQ=
chord PQ +tj, an infinitesimal
of at
least the thirdorder;
LM= BP-^-iy an infinitesimal of at least the firstorder;
TN^ RP"i\ an infinitesimal of at least the firstorder.
Hence, since the chord PQ
=v^Ax, it follows that
^ surface generated by arc PQ
i+(g)'..+,)
Therefore,
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154 INTEGRAL CALCULUS [Ch. IX.
^ surface generated by AB
"
"=*! :
liiniW-0^2y + i\(\1+(f|Y-+
^)-
By a fundamental theorem* the least and the greatest expres-ions
in this inequality are each equal to
" = *!,
Hence, surface generated by AB
* = *!
=
Hmit:^=o^2ryyjl^f^'^.
When Ax approaches zero,-^
takes a definite limiting value,
dv^^
namely -^"Therefore, by the definition of a definite integral,
ax
area I of surface
=J^^*2"yVl(||)*^^'(1)
It is necessary to express the function under the sign of inte-ration
in terms of x before integration is performed. If
be used for the length of the chord, there will result,
"r"
area of surface
=J^^*2iry^H-(^)*"l2)
Formula (1)or formula (2)is taken according as it is more
convenient to choose x ov y for the independent variable.
The surface generated by the revolution of AB about the y-oxis
is
givenby the formulae,
* See Note D, Appendix.
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74.] FURTHEB GEOMETRICAL APPLICATIONS 166
s"rfaee=r2^Vl^^^-
(8)
(4)
The student is advised to deduce these formulae for himself.
The expressions under the sign of integration in formulae (1),
(2)may both be written 2wy
ds hy Art. 71, and those in formulae
(8),(4)may both be written 2 wx ds. In order to recall immedi-tely
a foi-mula for the area of a surface of revolution, it is only
necessary to remember that the area traced out by an infinitesi-al
arc in its revolution about any line is equal to the product of
the length of the infinitesimal arc by the length of the circle
which is described by a point on the arc.
Ex. 1. Find the surface generated by the revolution of a
semicircle ofradius a about its diameter.
Let the diameter be the x-axis,
and the origin be at the center ;
the equation of the curve will be
"" + y* = o*.
Surface generated by ABA'
about a;-axis
'^-rnMir*-But, 1 +
Hence,
""= 1 +fl;2
_
g^ + y'_
q'
surface = 2 xa j dx
Ex. d. Find the surface of the prolate spheroid obtained by revolving
about the oc-axisthe ellipse h^^-\-aV
= "^"^-
Ex., 3. Find the surface generated by revolving about the a;-axis the
parabola y*= 4
ax.Show
that the curved surface ofthe figure
generatedby the arc between the vertex and the latus rectum is 1.219 times the area
of its base.
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156 INTEGRAL CALCULUS [Ch. IX.
Ex. 4. Find the surface generated by revolving about the y-axis the
catenary y=-fe"-fe "]from x = 0 to x = a.
Ex. 6. Find the entire surface generated by revolving about the x-axis
the hypocycloid x' + y"
= a*
Ex. 6. A quadrant of a circle of radius a revolves about the tangent at
one extremity. Find the area of the curved surface generated.
Ex, 7. The cardioid r = a (1 + cos e) revolves ab"Tut the initial line.
Find the area of the surface generated.
75. Areas of surfaces whose equations liave the form z^fipc^y).
Areas of surfaces of revolution were considered in the last article.
A more general case will now be discussed. In the explanation
of the following method for measuring the area of a surface,
referencewill be made to these two geometrical
propositions:
(a) The area of the orthogonal projectionf a plane area upon
a second plane is equal to the area of the portion projectedulti-lied
by the cosine of the angle between the planes. (SeeC.
Smith, Solid Geometry, Art. 31.)
(b) If the equation of a surface be in the form z
=/(", y),the
cosine of the angle between the ipy-plane and the tangent plane at
any point {x,y, z)of the surface is
MtH%)V-(SeeC. Smith, Solid Geometry, Arts. 206, 26.)
Let z = f(x,y)be the equation of the surface BFCMALB
whose area is required. Take two points P, Q,whose coordinates
are x, y,z,X'{- Ax, y -{-Ay, z
-{-Az, respectively. Through P and
Q pass planes parallel to the ^2;-plane and let them intersect the
surface in the arcs ML, M^Ly Also pass planes through P, Q,
parallel to the gaj-plane. The curvilinear figure PQ is thus
formed. Theprojectionf the surface PQ on the i"y-plane is the
rectangle PiQi whose area is Ax Ay. When Ax, Ay approach zero,
the point Q comes infinitely close to P; and the curvilinear sur-
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74-76.] FURTHER GEOMETRICAL APPLICATIONS 157
face PQ, which is then infinitesimal, approaches coincidence with
thai portion of the tangent plane at P, which also has PxQi ior its
projectionn the a^-plane. The area of PjQi also becomes dxdy.
Fio. 46.
Now let Q be infinitelynear to P. Ify
is the angle between the
a?y-plane and the tangent plane at P, it follows from (a)and the
remarks whichhave justbeen made, that
Hence,
area PiQi = area PQ " cos y.
area PQ = area PjQi " sec y
= dx dy sec y.
Therefore, by (6),area PQ =
yjl^f^\f^Yx dy.
The summation of all the infinitesimal surfaces PQ in the strip
LMMiLi gives
area of strip LM. =
[/ i+ (:")X^hr
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158 INTEOBAL CALCULUS [Ch. IX.
The summation of all the strips like LM^ in the surface
BFCMALB gives
area"
" " --"-- -
CSS 01 f^SL I
, of sxMace BFCMALB=C
[ C
\^+(^'+(j-)
or, abbreviating in the usual way,
The limits y = SL, x = OA can be determined from the equa-ion
of the surface. It is necessary to express the function under
the signs of integration in terms of x and y. It may happen that
a more convenient form of the equation of the surface is either
x = f(y,z),
r y
=f(z,x).
The area of the surface will then be
the value of either one or the other of the double integrals
between the proper limits of integration.
In some cases,
thereare two
surfaces each of whichintercepts
a portion of the other. In finding the area of the intercepted
portion of one of the surfaces, it is necessary to obtain the partial
derivatives that are required in the formulae of integration, from
the equation of the surface whose partial area is being sought.
This is illustrated in Ex. 2.
Ex. 1. Find the surface of the sphere whose equation is
xs + ys + "2 = a^.
Let 0-ABC (Fig.46) be one eighth of the sphere. In this case,
dx z' dy "*
\dx) \dy/ z^ z^ z^ a2-x2-ya
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75.] FURTHER GEOMETRICAL APPLICATIONS 159
XT^OA y=8L
Therefore, area of surface ^5C = J J ^i+l^\\l^y(ix
Jo Jo Va^ - X2 _ y2
^0 L Vo^ " x-'-^o
2 Jo 2
Hence, area of all the surface of the sphere is 4 ra*. (Compare Ex. 1,
Art. 74.)
Ex. 2. The center of a sphere, whose radius is a, is on the surface of a
right cylinder the radius of whose base is i a. Find the surface of the cylin-er
intercepted by the sphere. On taking the origin at the center of the
sphere, an element of the cylinder for the 2r-axisand a diameter of a right
section of the cylinder for the a"-axis,the equation of the sphere will be
and the equation of the cylinder, x^ + t/^ ax.
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160 INTEGRAL CALCULUS [Ch. IX.
The area of the strip GP will first be found, and then the strips in the
cylindrical surface APBOCA will be summed. The element of surface in
the strip CP iadzdz. Hence,
Cylindrical surface intercepted = 4 APBOCA
Since the surface required is on the cylinder, the partial derivatives must
be derived from the second of the equations above. Hence,
dy_a-2x
djL^Q
dx 2y'
dz
Also, CP^ = "a = a* " (x + y2),since P is on the sphere,
and hence, =a^ " ax, since P is on the cylinder.
Moreover, OA = a.
Therefore,the cylindrical surface
intercepted
-if L'-(^")']**-But on the cylinder, y'^= ax " x^. Hence,
the intercepted cylindrical surface
^"'-'"
dxdz
=2a((
^0 Jo Vox - x*
=
2ar^^^^dx= 2arJ^-dx
"^0 Vax - x2 -^0^x
= 4a2.
Ex. 3. In thepreceding example,
find thesurface of
thesphere
inter-epted
by the cylinder.
Ex. 4. Find the area of the portion of the surface of the sphere
x2 + y2 + 2ra- 2 ay
lying within the paraboloid
y = Ax^-\- BzK
76. Mean values. The mean value of n quantities is the nth
part of their sum. Let 4^(x)be any continuous function of a?,
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76-76.] FURTHER GEOMETRICAL APPLICATIONS 161
and let an interval, 6 " a, be divided into n parts, each equal to h.
The mean value of the n quantities,
"^(a),^(a+ ^),it"{a 2h),
...,
"^(a+ (n- 1)^),
"\"{a)'\-"f"{a-\-h)'\-"\"{a-\-2h)-\"^(a4-n 1^)
n
Since n = "7^,this mean value is
6 " a
Now suppose that x takes all the possible values, infinite in
number, that are in the interval between a and 6. Then, n is
infinite,h is infinitesimal, and the number of terms in the last
numerator is infinite. The sum of all these terms, by Art. 4, is
expressed by
fj"(x)dx.
Hence, the mean value of all the values that a continuous
function, ^(a?),an take in the interval 6 " a for a? is
J^4"(ag)clag
This is usually called the mean value of the function"\"(x)
ver
the range b " a. A geometric conception of the mean value was
given in Art. 7 (a). A more general definition of mean value is
given in Art. 77.
It is necessary to understand clearly the law according to
which the successive values of the function are taken. Exs. 1, 2,
Exs. 6, 7, and Exs. 12, 13, will serve to illustrate this remark.
Ex. 1. Find the mean velocity of a body when falling from rest, the
velocitiesbeing taken at
equalintervals
oftime.
In the case of a body falling from rest, v = igt. Hence, calling V the
mean velocity for a time tu
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162 INTEGRAL CALCULUS [Ch.IX.
\vdt
that is, the mean velocity is one half the final velocity.
Ex. 8. In the case of a body that falls from rest, find the mean of the
velocities which the body has at equal intervals of space. It is known that,
if 8 is the distance through which a body has fallen on starting from rest,
and V is the velocity required.
Hence, the mean velocity,
that is, the mean of the equal-distance velocities is equal to two thirds of the
finalVelocity..
Ex. 8. Findthe average value of the
function3a;3 +
5x "
7as x varies
continuously from 1 to 4.
Ex. 4. Find the average value of the function "^ " 3aj" + 2a;^l ass
varies continuously from 0 to 3.
Ex. 5. Find the average ordinate drawn,
(o) in the curve, y = x^ + x + 1 between the abscissas 2, 3 ;
(6) in the curve, y = (x+ l)(x + 2) between the abscissas 1, 3 ;
(c) in the curve, x* + ox* + cM^ + ft^y= o between the abscissas o, 0.
Ex. 6. Find the mean length of the ordinates of a semicircle (radiusa),the ordinates being erected at equidistant intervals on the diameter.
Ex. 7. Find the mean length of the ordinates of a semicircle (radiusa),the ordinates being drawn at equidistant intervals on the arc.
Ex. 8. Find the mean value of sin ^ as ^ varies from 0 to -"
2
Ex. 9. Find the mean distance of the points on the circumference of a
circle of radius a, from a fixed point on the circumference.
Ex. 10. Find the mean latitude of all the places north of the equator.
Ex. 11. A number n is divided at random into two parts. Find the mean
value of their product.
Ex. 12. Show that the mean
ofthe
squareson the diameters
ofan
ellipsethat are drawn at equal angular intervals is equal to the rectangle contained
by the major and minor axes.
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76-77.] FURTBEB GEOMETRICAL APPLICATIONS 163
Ex. 18. Show that the mean of the squares on the diameters of an ellipse
that are drawn at points on the curve whose eccentric angles differ succes"
sively by equal amounts, is equal to one half the sum of the squares on the
major and minor axes.
77. A more general definition of mean value. In Exs. 1, 3, 4,
below, " the range"
over which the function (inthese cases, the
distance of a point)varies,is a plane area. In Ex. 2, the range
is a curvilinear area ; and in Exs. 5, 6, it is a portion of space.
The following may be taken for the definition of the mean value
of a function, whatever the range may be :.
The mean value of a
fimction throughout
any range
(The value of the ftinctionfor eaeh
element of the range) x (the ele-
.mentof the range)
The range
in which the summation in the numerator is made throughout
the whole of the range. The mean value considered in Art. 76
is merely a special case.
Ex. 1. Find the mean distance of a fixed point on the circumference of a
circleof radius a from all points within the circle.
On taking the fixed point for the pole and the tangent thereat for the
initial line, the value of the function (inthis case the distance)t any point
(r,e) is r. The element of the range
(inthis case an area)at the point is
rdedr. This is shown in Fig. 47.
Hence,
the mean distance =
i I r^drde
_
Jo Jo
Jo Jordrde
8a8
^
1 sin8 e'de3 Jo
9ir
(SeeEx. 1, Art. 67.)
INTEGBAL CALC. " 12
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164 INTEGRAL CALCULUS [Ch. IX.
Ex. 8. Find the mean length of the ordinates drawn from all points on
the curved surface of a hemisphere of radius a to itsdiametral plane.
Ex. 8. Find the mean length of the ordinates drawn from all the points of
its diametral plane to the surface of the hemisphere of radius a.
Ex. 4. Find the mean square of the distance of a point within a given
square (side 2a)
from the center of the square.
Ex. 5. Fmd the mean distance of all the points within a sphere of radius a
from a given point on the surface.
Ex. 6. Find the mean distanceof alltbe points within
a
sphere of radiusa
from the center.
EXAMPLES ON CHAPTER IX.
1. Find the volume of a sphere of radius a by means of a single integra-ion.
(Suppose that the sphere is made up of infinitely thin concentric
spherical shells of thickness dr. The volume of each shell = 4TrMr ; hence
volume of sphere = 4 ir I n^dr = J wa^.
Jo
8. Find the volume and surface generated by revolving about the y-axis
the ellipse 6%2 + ^2^2-
^252,
8. Find the surface generated by the revolution about the ysuasoi the arc
of the parabola y^ = 4ax from the origin to the point (",y),
8 a?4. Find the volume generated by revolving the witch y = " " " - about
its asymptote.^
6. Find the convex surface of the cone generated by revolving about the
ic-axisthe line joiningthe origin and the point (a,b).
6. Find the surface of the torus generated by revolving about the a;-axis
the circle x^ +(y - 6)2 = a*.
7. On the double ordinates of the ellipse ^-f^= 1, and in planes per-
a2 52
pendicular to that of the ellipse, isosceles triangles of vertical angle 2 a are
described. Find the volume of the surface thus constructed.
8. Two cylinders of equal altitude h have a circle of radius a for their
common upper base. Their lower bases are tangent to each other. Find the
volume common to the two cylinders.
9. Find the volume inclosed by two right circular cylinders of equal
radius a whose axes intersect at right angles. Alaa^ find the. surface of one
intercepted by the other.
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77.] FUBTHER GEOMETRICAL APPLICATIONS 165
10. Find the volume of the solid contained between
the paraboloid of revolution, x^
-\-ff^^az)
the cylinder, "? + y^ = 2 ax ;
and the plane, ;? = 0.
11. Find the entire volume bounded by the surface
"' + y* 4- "* = o*
12. An arc of a circle revolves about its chord. Find the volume and sur-face
of the solid generated, a being the radius, and 2 a the angular measure of
the arc.
18. A cycloid revolves about the tangent at the vertex. Find the surface
and volume of the solid generated.
14. A cycloid revolves about its base. Find the area of the surface
generated.
15. A cycloid revolves about its axis. Find the surface and volume
generated.
16. A quadrant of an ellipse revolves round a tangent at the end of the
minor axis of the ellipse. Find the volume of the solid generated.
17. If 6 be the radius of the middle section of a cask, a the radius of
either end, and h itslength, find the volume of the cask, assuming that the
generating curve is an arc of a parabola.
18. Find the length of the curve ^aif^ z{x" Z
o)2from a; = 0tox = 3a.
19. Find the length of the logarithmic curve y = ca*.
20. Find the length of an arch of the epicycloid,
X =
(o+
6)cos^ - 6
cos^^-i-^,
h
y = (o + 6)sin^ - 6sin^^-"-^^.b
21. Find the length of an arc of the evolute of the parabola y^ = 4 px,
namely,
27py3 = 4(x-2p)"
from the
pointwhere x = 2p to the
pointwhere x = 3p. Also, find the
length of arc of the preceding curve from the cusp (x = 2/)) to the point
where it intersects the parabola (atthe point for which x = 8j)).
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166 INTEGRAL CALCULUS [Ch.IX.
32. Find the length of the arc of the curve y = logsinx between a; =
3
and 05 = -"
2
88. Find the length of the arc of the evolute of the circle,
a5 =
o(co8^ + ^sin^),
y =
o(8in^" ^co8^),
from ^ = 0 to ^ = a.
a
94. Find the entire length of the curve r = asin^-.
26. Find the length of arc of the spiral r = m^, from ^ = 0 to ^ = ^i.
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CHAPTER X
APPLICATIONS TO MECHANICS
78. Mass and density. This chapter is introduced for the
purpose of giving the student further examples of the applica-ion
of the fundamental principle of the integral calculus, and
of affording him additional opportunity for practice in integra-ion.
The definitions of mechanics that are required in what
follows are merely stated, but are not discussed. They will be
familiar to thosewho
have had theadvantage of
an
elementarycourse in that subject.
Other readers can only assume these
definitions as data for problems in integration.
Ma^s, The mass of a body is usually defined as" the quantity
of matter which it contains," and is specified in terms of the
mass of a standard body. In English-speaking countries, for
ordinary purposes, the
standard
mass is a certain bar ofplati-um
marked "P.S. 1844. lib.," which is called the "imperial
standard pound avoirdupois," and is preserved at the Office of
the Exchequer in London. Any mass equal to this standard
mass is then a unit of mass. For scientific purposes in general,
and in countries where the metric system is adopted, the standard
of mass is the "kilogramme des archives," a bar of platinum
kept in the Palais des Archives in Paris. A mass equal to one
thousandth of this standard is then the unit of mass ; this unit
is called the gram. The mass of a body should not be con-founded
with its weight. The weight of a body depends upon
its distance from the center of the earth, but its mass is inde-endent
of its position.
Density. The mean density of a body is the quotient of its
mass by its volume. The density at a given point of a body is
167
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78-79.] APPLICATIONS TO MECHANICS 169
smaller the portions Am the more nearly do they come to being
particles with distances rj, r^ """, from the fixed plane. Ulti-ately,
therefore, the distance of their center of mass from the
plane is given by
In accordance with the first definition of integration this is
written,
I rdm
Jdm
Therefore, in the case of any continuous distribution of matter,
the coordinates x, y, z, of the center of mass are given by
ixdm \ydm \zdtn(4)
\ dm \ dm \ dm,
Ifp
be the density at any point of a body, and dv an infini-esimal
volume about the point,
dm^^pdVf
the total mass =
Jp dv^
and formulae (4)become
_
\^dv_
\pydv_
iftzdv
^ = ^ ; y--^ ; z = ^ " (5)
\^dv \^dv \^dv
The densityp usually varies
frompoint
topoint of
a body,and
it is generally expressed as some function of the position of the
point. If the body is homogeneous,p
is constant and can be
removed from formulae (5)by cancellation. If jobe the mean
density of a non-homogeneous body, then, by the definition in
Art. 78,
rpcfv
P=S (6)
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170 INTEGRAL CALCULUS [Ch. X.
If matter be supposed to be continuously distributed along a
line or curve making, as it were, a wire of infinitesimal cross-
section, or so thinly laid upon a surface, curvilinear or plane,
that the thickness of the layer may be neglected, the term "mass-
center"
can also be used with reference to lines and curves, sur-faces
and plane areas. If As, A^S',A^ are small elements of a
line or curve, a curvilinear surface, and a plane area respectively,
and pis the linear density or the surface density, the coordinates
of the mass-center of an arc, surface, or plane area are obtained
from formulae (5)on the substitution of ds, dS, dA respectively
for dV. Expressions for these differentials have already been
obtained in the preceding articles. The mean linear and surface
density can be obtained by making these substitutions in for-ula
(6).
Ex. 1. Find the total mass and the mean density of a very thin plate
which is the firstquadrant of the circle whose equation is x^"\-1/^
a^, and
whose density varies at each point as xy.
If p denote the density, then by the given
condition,
pccxy;
that is, p = kxyj
in which k is some constant.
If M denote the total mass of the quadrant,
and dm denote the mass of an infinitesimal rec-tangle
about any point,FiQ. 48.
M=^(dm=(pdA
= r('^'''"^kxydxdyJo Jo
= ika*.
If p be the mean density, it follows from the definition that,
(pdA
Jira2~2ir
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79.] APPLICATIONS TO MECHANICS 171
Here,
Ex. 8. Find the center of mass of the thin plate described in Ex. 1.
ipxdv ikxyxdA
(pdv M
M
Similariy, y = A ""
Ex. 8. Find the mass-center for a thin hemispherical shell,radius a, whose
density at each point of the surface varies as the distance y from the plane of
the rim.
Let the hemisphere be described by revolving tlie semicircle of radius a
and center O about the y-axis O F, which is at right angles to the diameter,
the point 0 being takenfor
the origin
of coordinates. Let P, whose coordi-ates
are ", y, be any point on the
semicircle, and draw PJtf, PN at
right angles to the axes of x and y
respectively. Join OP^ and denote
the angle NOP by e.
At the point P, y = a cos ^ ;
also at P, p X y,
tliatis, p" ka cos 0,
in which k is some constant. The infinitesimal arc of length ds at P describes
a zone about OF whose area is given by
d8 = 2ifNPd8.
or, since ds = add, = 2 ira sin ^ . o dtf.
The symmetry of the figure shows that
x = 0.
rf
Also,
(pydS 2nka*
f'^^
cos^ 0 sin 0 d0
y =l ^
XpdS2vka^
i^ cos 08iR0d0
Hence, the center of mass is at the point (0,| a).
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172 INTEGRAL CALCULUS [Ch. X.
Hence,
Ex. 4. Find the center of mass of a lig^tcircular cone of height A, which
is generated by the revolution of the
line y =: ax about thex-axis, when
the density of each infinitely thin
croBB-section varies as its distance
from the vertex.
Symmetry shows that the center
of mass is in the z-axis. Suppose
that a very thin plate B8 is taken
which cuts the axis of the cone at
right angles at C at a distance x
from the vertex.
The radius CB of the cross-section = ax.
The density of this thin plate, p = kx.
The volume of the thin plate, dV=ir cS^ dx
Ex. 5. Find the mean density of the cone described in Ex. 4.
Ex. 6. Find the mass-center of the surface of the cone in Ex. 4.
Ex. 7. Find the mass-center of the cone generated in Ex. 4, and the mass-
center of its convex surface when the density is uniform.
Ex. 8. Find the mass-center of a quadrantal arc of the hypocycloid
asf-f y" = a'.
Ex. 9. Find the mass-center of the convex surface of a hemisphere of
radius 10.
Ex. 10. The quadrai\t of a circle of radius a revolves about the tangent
at one extremity ; prove that the distance of the mass-center of the generated
curved surface from the vertex is.876
a.
Ex. 11. Find the mass-center of the semicircle of x* + y^ = a2 on the
right of the y-axis.
Ex. 18. Find the mass, the mean density, and the mass-center of the
semicircle in Ex. 11 when the density varies as the distance from the
diameter.
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79-80.] APPLICATIONS TO MECHANICS 173
Ex. 18. Find the mass-center of a circular sector of angle 2 a, taking
the origin at the center, and the x-axis along the bisector of the angle.
Ex. 14. Find the mass-center of the first quadrant of the ellipse b^^.
+ a2y2 = a^b^.
Ex. 16. Find the mass-center of the area between the parabola y^ = 4 az^
and : (a)the double ordinate for a; = ^ ; (b)the ordinate ioTx = h and ac-axis.
Ex. 16. Show that the mass-center of the circular spandril formed by a
quadrant of a circle of radius a and the tangents at its extremities is at a
distance
.2234
a from either tangent.
Ex. 17. Find the mass-center of a quadrant of the hypocycloid "" + y*
= a*.
Ex. 18. Find the mass-center of the area between the parabola x* + y*
= a^ and the axes.
Ex. 19. Find the mass-center of the area between the cissoid y^ =-^
"
and its asymptote.a
"
x
Ex. 20. Find the mass-center of the cardioid r = 2 a (1 " cos ^).
Ex. 81. Find the center of mass of the solid paraboloid generated by the
revolution of y^ = 4iax about the ic-axis.
Ex. 22. Show that the center of mass of a solid hemisphere of uniform
density and radius a, is at a distance | a from the plane of the base.
Ex. 28. Show that the center of mass of a solid hemisphere, radius a, in
which the density varies as the distance from the diametral plane is at a dis-ance
^ a from this plane. Also show that the mean density of this hemi-phere
is equal to the density at a distance f a from the base.
Ex. 24. Find the center of mass of a solid hemisphere, radius o, in which
the density varies as the distance from the center of the sphere.
Ex. 26. Find the center of mass of the solid generated by the revolution
of the cardioid r = 2 a (1 "
cos^Jaboutitsaxis.
80. Moment of inertia. Radius of gyration. If in any system
of particles the mass of each particle be multiplied by the square
of its distance from a given line; the sum of the products thus
obtained is called the moment of inertia of the system about that
line. Thus, if m^, m^, """, be the masses of the several particles,
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174 INTEGRAL CALCULUS [Ch. X.
**!" ""2" *"} ^^eir distances from the line, and I denote the moment
of inertia,
/= miVi^-f
wigrs*-I-
"" ;
that is, /= l,m7^. (1)
In any case in which matter is continuously distributed, as in
a solid cylinder, a shell, etc., the matter may be supposed to be
divided into small portions, A??ii, Awij, """. By reasoning similar
to that employed in the last article,it can be shown that
If matter be supposed to be distributed uniformly along a line
or curve, or upon a curvilinear surface or a plane area, the term
"moment of inertia" can also be used in reference to curves,
surfaces, and plane areas.
Let M denote the total mass of a body, namely I dm, and I its
moment of inertia about a given line or axis. If k satisfies.the
equation
r1 r^dtn
that is,if A;2 = ^ =^
,
^
jam
k iscalled the radius of gyration of the
bodyabout the given
axis.
Ex. 1. Find the moment of inertia of a rectangle of uniform density,
whose sides have the lengths ft, d about a line which passes through the
center of the rectangle and is parallel to the sides of length ft.
The density per unit of area will be represented by unity. Let the axes
of X and y be taken parallel to the sides of the rectangle, the origin being at
the center, and let
AB^h, BC = d.
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80.] APPLICATIONS TO MECHANICS 175
Then 1= (y^dA
d h
= j'j y^dydx
12*
This moment of inertia is impor-ant
in calculations on beams. Since
the mass of the i*ectangle = "d,
M 12*
*" = ^ =
^
dx
Udy
Fig. 51.
Ex. 8. Find the moment of inertia of a very thin circular plate of uniform
density of radius a about an axis through its center and perpendicular to
its plane.
Taking the density as unity per unit of area,
"Jo Jo r'^^rdrde
Also,
2
*
TO*
lc =L
=l^
=^.
M Tcd^ 2
Ex. 8. Find the moment of inertia about itsaxis of a right circular cone
of height h and base of radius 5, the density being uniform, and m being
the mass per unit of volume.
The moment of inertia is equal
to the sum of the moments of inertia
of very thin transverse plates like
B8, If
OC:=X,
then, by similar triangles.
Fig. 62.
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176 INTEGRAL CALCULUS [Ch. X.
Hence, if dl denote the moment of inertia of the plate B8 of thickness
(to,by Ex. 2,
Therefore, for the whole cone.
^X"""-'"to
10
Also, i^^L^J-^iLB^I^M mV \mrcl^h
Ex. 4. Find the radius of gyration of a uniform circular wire about its
diameter.
Ex. 6. Find the moment of inertia of the triangle formed by the axes and
a line whose intercepts are a and 6, about an axis which passes through the
origin, and is at right angles to the plane of the triangle.
Ex. 6. Find the radius of gyration about its line of symmetry of an
isosceles triangle of base 2 a and altitude h.
Ex. 7. Find the moment of inertia about the x-axis of the area between
the line and the parabola which both pass, through the origin and the point
(a,6),the axis of the parabola being along the x-axis.
Ex. 8. Find the moments of inertia of the ellipseh'h? + aV = a^'^ : (a)
about the x-axis ; (6)about the ^-axis ; (c)about an axis that passes.through
the center of the ellipse and is perpendicular to the plane of the ellipse.
Apply the results to the circle x^ + y^ = a^.
Ex. 9. Find the moment of inertia of the thin plate in Ex. 1, Art. 79,
about the x-axis.
Ex. 10. Find the moment of inertia of a homogeneous ellipsoid
aboutthe
x-axis.
Ex. 11. Find the moment of inertia of the surface of a sphere of radius a
about a diameter, m being the mass per unit of surface.
Ex. 12. Find the moment of inertia of a solid homogeneous sphere of
radius a about a diameter, m being the mass per unit of volume.
Ex. 18. Find the moment of inertia of the semicircular plate described in
Ex. 12, Art. 79, about the diameter.
Ex. 14. Find the moment of inertia, and the radius of gyration about its
axis, of a homogeneous right circular cylinder of length I and radius JB, m
being the mass per unit of volume. Also about a diameter of one end;
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CHAPTER XI
APPROXIMATE INTEGRATION. INTEGRATION BY
MEANS OF SERIES. INTEGRATION BY MEANS
OF THE MEASUREMENT OF AREAS
81. Approximate integration. It was remarked in Arts. 4, 8
that in most cases in which a differential f(x)'dxis given it is
not possible to find the anti-differential. In some of these
cases, however, an expression can be found that will approxi-ately
represent the indefinite integral
jf(x)x. Even if this
cannot be done, it is often possible to determine a value that
will very nearly be that of the definite integral jf(x)dx.
Art. 82 explains a method, that of integration in series, by
means of which an indefinite integral may be expressed as a
function of x in the form of a series that contains an infinite
number of terms. An important application of this method
to another problem is given in Art. 83. Arts. 84--87 set forth
a method, that of measurement of areas, which reduces the
evaluation of a definite integral to a mere matter of careful
computation. In this connection several formulae for the ap-roxima
determination of areas are necessarily considered.
82. Integration in series. When the indefinite integral of a
given functiouj f(x)dx,cannot be found by any of the means
thus far considered, one of the most usual and most fruitful
methods employed is the following: The function f(x) is de-eloped
in a series in ascending or descending powers of x. If
this series is convergent within certain limits for x, the series
obtained by integrating it term by term is also convergent
177
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178 INTEGRAL CALCULUS [Ch. XI.
within the.same
limits.* The greater the number of terms
taken the more nearly will the new series representif(x)dx,
Ex. 1. ""^'**'' *1. Find f-
(1 + a*)*
By the binomial theorem,
(l+ x^)*1 3^1.2 32 1.2.3 3"^
"
The second member is convergent for valaes of x between + 1 and " 1.
Integration of both members of (1)gives
^^^,2
_"L4.?_L^
g^^ 2.6.8 g^
(1+ ic^)*1
'
3 . 6 1 . 2
*
32 . 11 1 . 2 . 3'
3" . 16(2) J
The second member represents the required integral for values of x
between 4- 1 and " 1. It foUowafrom (2)that
/:dx
^
..2 2.6 1 2.6.8 1
(l+x*)*3.6'^1.2'38.11 1.2.3'3".16'^
Ex. 2. Find (e=^dx.
Since e" = 1 + " +1.21.2.3
(1) e"'=l + .^ +
j^+ ^^+.
which is convergent for all finitevalues of x.
* Suppose that
f(x) = ao + aix + 02*2 + ... + an-ix"-i + a^x" + .... (1)
Then Cf(x)dxa^ +
^+^+^..+^^i^=^
+ ^^^^^ (2)^ 23 "n+l
The series in (1)is convergent when-^^
is less than unity for all values
of n beyond some finite number. The series in (2) is convergent when
" - "
.^i^^and therefore when
^^, is less than unity for all values of n
n + 1 a"-i a"_i
beyond a certain number. Since theeonvergency of
bothseries
depends
upon the same condition, the second series is convergent when the firstis
convergent.
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82-83.] APPROXIMATE INTEGRATION 179
Integration of both members of (1)gives
2.0 1.2.3.71.2.3.4.9
Ex.
(2) j'e''dxc +
a;+^+-The second member represents the required integral for values of x
between 4- 1 and " 1. It follows from (2)that
J-i V 3^1.2.6
1.2.3.7
1.2 .3.4.9 /
Ex 4 f ^ ^^- *" f"==" (Putsina;=;".)
^ ^
Ex. 9. f?!^.Ex.6. \x^\/\-v:^dx, "^
*
"^Ex.10. J^ to.
Ex. 6. fa;2Vl-iC'^dx.
"^ Ex.11. f?!I^(to.(Compare Ex. 28, page 98.) J x
83. Expansion of functions by means of integration in series. A
function can be developed in series by means of the method
described in the last article if the expansion of its derivative is
known. The series which represents the function is obtained by
integrating the series which represents the derivative, and deter-ining
the value of the constant of integration.
Ex. 1. Expand tan"i a: in a series of ascending powers of x.
Differentiation and division give
d. tan-isc =
-^= (1 - a;2 + "" - jc" + ... + (- l)'Hc2* )dx,
I + a:*
which is convergent when x liesbetween " 1 and +1.
Integi'ating,
tan-ix = c +
x-^+^-?I+...
+ (-l)n"*"^'
--
3 6 7^ '
2n+l
The substitution of 0 for x gives
mv = c,
m being an integer ; and hence,
tan-la = mT +a;-^H-^-?^+
...
3 6 7
INTEGRAL CALC. " 13
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180 INTEGRAL CALCULUS [Ch. XI.
This series* can be employed for values of x between " 1 and + 1. It
can be used for computing the value of v. For, on putting x = " therein,
it is found that "^
tan-" J-=m"- + 5:
= "",r +J- f1 -
i+i
-
-I-+ ... V
y/S" VS\ 0 46 189 /'
whence, , = 2v/3 (ll +"- J5+
...).Ex. 2. Expand t sin-^ a; in a series in x ; and compute the value of v by
putting x = J.
Ex.8. Derivet
j'e-.'dxl-^g ^^-j-^+....hich ta
convergent for all finitevalues of x.
Ex.4. Show that
log(a + x)=loga + ^-^,3^-^^-f...when|
and that log (a + X) = log a + ^-
-^+ ^ -
-^+ . " . when |a; |" 1.
x 2x^ oac* 4x*
The symbol |x \denotes the absolute value of x.
Ex. 5. Derive series for log (1 4- x),log (1 -
x),log 2, log 9.
Ex. 6. Develop log (x + Vl + x^)in a series by integrating (1+ x*)"^dx.
Ex. 7. " Show that
k^ being less than unity. (SeeEx. 9, Art. 46.)
* It is usually called Gregory's series, after its discoverer, James Gregory
(1638-1676). It was found also by Leibniz (1646-1716).
t This expansion is due to Newton (1642-1727),nd, by means of it,he
computed the value of ir.
t This integral is often met in the theory of probabilities, and in certain
questions in physics. For the evaluation of t "-**c?x when x is greater than
unity, see Laurent, Cours d*Analyse, t. III., " IV., p. 284. For the deriva-
tion of (e-'^dx = J Vir,see Williamson, Integral Calculus, Ex. 4, Art. 116.
" This integral is called the **
elliptic integral of the first kind." It re-ceived
the name ellipticintegral from its similarity to the integral in Ex. 8,
which represents the length of a quadrantal arc of an ellipse, and is known
as "the
ellipticintegral
ofthe
second
kind." The integralof
the first
andsecond kind are usually denoted by F(Jc,^),E(Jc,0), respectively. These
names and symbols were given by Legendre (1762-1833).
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83-84.] APPROXIMATE INTEGRATION 181
Ex. 8. Show that
(See Ex. 9, Art. 46 ; Ex. 7, Art. 71.)
Ex. 9. Deduce the value of t by means of the series in Ex. 1, itbeing
known that - = 4 tan- 1i tan- 1
-^.5 239
84. Evaluation of definite integrals by the measurement of areas.
It has been seen in Arts 4, 6, that the definite integral I f(x)dx,
may be graphically represented by the area included by the curve
whose equation isyz=f(x),the axis of x, and the ordinates for
whichx = ay x = b] and it has been
observed
that the
evaluation
of the integral is equivalent to the measurement of this area.
The numerical value of the integral j f(x)dx, which is also the
same as that of the area justdescribed, has been obtained up to
this point, by finding the anti-differentialof f(x)dXf say "t"(x),
substituting b and a for x therein, and calculating " (6) "^(a).
But when it is not possible to find the anti-differentialof f(x)dx,
recourse must be had to other methods.
While, on the one hand, as already shown, areas may be
determined by evaluating definite integrals, on the other hand,
definite integrals may be evaluated by measuring areas. If the
anti-differentialof f{x)dxis unknown, the value of j f(x)dxcan
be found in the following way. Plot the curve y=zf(x)from
x=:a to x = b, erect the ordinates for which x = a, a; = 6, and
measure the area bounded by the curve, the axis of x, and these
ordinates. There are several rules or formulae for determining
areas of this kind. The degree of approximation to absolute
correctness depends in general only on the patience of the
calculator. These formulae, some of which are usually given in
manuals for engineers, are called ^'formulae for the approximate
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182 INTEGRAL CALCULUS [Ch.XI.
determination of areas," or "formulae for approximate quadra-ure."
They may be given the more general title, ^^formulce
for approximate integration.^^ The two rules most frequently
employed, namely, the trapezoidal rule and Simpson's one-third
rule, are discussed in Arts. 85, 86, and a rule deduced from them
is given in Art. 87. Other rules are given in the Appendix.*
It should be observed that only a numerical result is obtained
by means of these rules. The knowledge of the value of the
definite integral | f{x)dx thus calculated does not give any clue
whatever to the expression of the indefinite integral I f{x)dx as
a function of a?. If the indefinite integral \f{x)dx has been
found in the form of a series which is convergent for values of x
between a and 6, the value of the definite integral I f(x)dXy
can be found as accurately as one pleases by taking a suffi-iently
large number of terms. Illustrations of this remark
have been given in Exs. 1, 2, Art. 82, and in Exs. 1, 2, 7, 8,
Art. 83.
85. The trapezoidal rule. Let AK^ be a portion of a curve whose
equation may or may not be known ; and let LA^ TK, be drawn
at right angles to the line
OX, It is required to find
the area AKTL contained
between the curve AK, the
line XT, and the perpendicu-ars
LA, KT,
Divide LT into n parts,
each equal to h, and at the
points of division erect the
perpendiculars MB, NC, """,
8H, Draw the chords AB,
BO, """, HK. A
rule
for finding the area
of
LAKT
will
now be
" See Note E.
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184 INTEGRAL CALCULUS [Ch.XI.
Ex. 2. Show that the approximate value obtained for'the above integral,
by making 20 equal intervals, is 333}.
ru
Ex. 8. Show that the approximate value of I \ogioxdx, unit intervals
being taken, is 7.990231.
86i The parabolic or Simpson's *one-third rule. The parabolic
rule for approximating to the value of the area LAKT is derived
by substituting parabolic arcs through ABCy CDE, """, GHK, for
thearcs
of the givencurve
passing through these points, theaxes of the parabolas being vertical, and then summing the
areas of the parabolic sections, LABGN, NGDEP, ..., RGHKT,
Fio. 64.
which are thus formed. A parabolic arc, as CDE, will more
nearly coincide with the given curve through CDE, than will
the chords CD, DE. For the purposesof
thisrule,
n the num-
berof equal parts into which LT \^ divided must be even, since
a parabolic strip is substituted for each of the consecutive pairs
of trapezoidal strips ; for example, NCDEP for ND-^
DP.
The area of one of the parabolic strips, say NCDEP will first
be found. Through D draw CE^ parallel to the chord CE, and
produce NC, PE to meet C'E^ in C, E\
" Thomas Simpson (1710-1761).
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85-86.] APPROXIMATE INTEGRATION 186
The parabolic strip NCDEP=: trapezoid NCEP+ parabolic
segment CDE,
The parabolic segment CDE = two thirds of its circumscribing
parallelogram CCE'E,
Hence, the parabolic strip
NCDEP= NP^NC+PE) + "{QZ" - |(J\r(7+^ j]
^2h(:^N0-\-iQD-^iPE)
=hNO
+ ^QD-\-PE).o
Application of the latter formula to each of the parabolic strips
in order beginning with the first on the left,and addition, gives,
approximately,
area LAKT = ^a + 4 + 2 + 4 + 2 + . -. + 2 + 4 + 1),8
in which merely the coefficients of the successive perpendiculars
LA, MB, ..., TK are written. As in the case of the trapezoidal
rule, the greater the number of equal parts into which iT is
divided, the more nearly equal will the area thus calculated be
to the true area.
If the equation of the curve AK isy=f(x), and OL = a,
OT=b, and XT is divided into n parts, each equal to"~ ^,
then
lengths of the successive ordinates, LA, MB, ."" TK, are /(a),
ffa+^-^=^\fa-h2 ^^^\ ..., f(b). Hence, on calling these
successive lengths, yo, yi, t/a """ yn"
rf(x)cb:^(yo-\-4.y,-\-2y,^4.y,-\-2y..
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186 INTEGRAL CALCULUS [Ch. XI.
For the sake of computation, this may be put in the form,
.2b-avl
X|^/(x)"fx=|.^^[i(yoyn) + 2(yi + y8 + - + y"-i)
+ (ya + y4 + - +
yf"-2)]*/"lO
Ex. 1. Evaluate I ic* "to by this method, taking n = 10.
Here, yoi yu ^2* """, yio? are 0, 1, 81, 625,..., 10,000, respectively; and
hence, approximately,
rVda;=j"{J^Si^+2(l+81+625+2401+6661)(16+256+1296+4096)}= 20001f
The true value of the given integral is 20,000 ; thus the error is only IJin 20,000.
ru
Ex. 2. Show that the value of I logioxdx calculated by this rule for
n = 10, is 8.004704 (compare Ex. 3, Art. 85).
A comparison between these two rules is given in the following
quotation :t "The increase in accuracy (ofthe parabolic)ver
the trapezoidal rule is usually quite notable, unless the number
of ordinates become large, in which case they both approximate
more and more closely to the true value and to each other. In a
* If n be the number of equal intervals into which the range 6 " a is
divided, the outside limit of error that the parabolic formula for integration
can have, is
\ 2 ) 90n*
in which av is some value of x between a and 6, and f^ (x) denotes the
fourth derivative of /(as). The outside limit of error in the case of the
trapezoidal rule is
12 n2 -^^^^'
in which /"(x) denotes the second derivative of /("). If " is doubled, the
limit of error is reduced, therefore, to ^ and J of itsformer amount. (See
Boussinesq, Cours d^Analyse, t. II. 1, " 262, and Markoff, Differenzen-
rechnung, " 14, pp. 57, 59.)
t This is from an
article, entitled,**New Rules for Approximate Integrar-
tion,'* in the Engineering News (N. Y.),January 18, 1894, by Professor W.
F. Durand of Cornell University.
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86-87.] APPROXIMATE INTEGRATION 187
series of trials made by the author upon a number of integrals of
various forms for the purpose of testing the relative accuracy of
these rules, it was found for cases in which the locus was of single
curvature only that the trapezoidal rule required about double the
number of sections for equal accuracy with the parabolic rule.
Where the locus involves several changes of curvature, as in
lumpy and irregular curves, and the number of sections is moder-te,
one rule is as likely to be right as the other, and both are
likely to be considerably in error. For a large number of sec-tions,
however, the parabolic rule will show its superiority .as
above."
87. Durand's rule. From a discussion *on the trapezoidal and
parabolic rules. Professor Durand has deduced another rule for
which" it seems not unfair to claim substantially the full prob-ble
accuracy of the parabolic rule, and practically the simplicity
in use of the trapezoidal rule." It is as follows, merely the co-effici
of the successive ordinates being written in order from
the left:
approximate area= ^[3%-|-|f
1 4-1 + """ +1-hl4-}f
+1^];
or,
approximately,
area = 7^[.4+ 1.1 + 1 + 1 f """ + 1-f-
1 + 1.1 +.4].
The number of intervals may be even or odd.
reo"
Ex. 1. Find the value of I sin 6 d$ with 10" intervals.Jo
The circular measure of 10" is.17453.
The rule gives for the approximate
value of the integral,
J sin ede =
.17453[.4(sin" + sin 60")+ 1.1 (sin10" + sin 60")
^
+ (sin20" H- sin 30"-h sin 40")]=
.5000076.
reap r -iflo"Since the exact value of I sin ^d^ is " cos ^
,or
.6,the difference
between the above approximate and the true values of this integral is not
more than one
part
in 66,666.
* In the article mentioned in the preceding footnote.
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188 INTEGRAL CALCULUS [Ch. XI.
J'M)^ dx calculated by this rule with unit intervals gives0
a difference of one part in 3333.
r\%
Ex. 3. Show that \ logio x calculated by this rule vnth unit intervals is
8.004062. (Compare with Ex. 3, Art. 86, and Ex. 2, Art. 86.)
88. The planimeter. Attention has been drawn to the fact that
the value of a definite integral is also the value of a certain plane
area, and that, consequently, the measurement of the area is
equivalent to the evaluation of the integral. In Arts. ^6, 86, 87,
rules are given for approximately determining plane areas, and
other rules therefor are given in the Appendix.* These areas can
be measured exactly by instruments called mechanical integrators
or planimeters. A planimeter measures the area of any plane
figure by the passage of a tracer round about the perimeter of the
figure, the readings being given by a self-recording apparatus.
There are several kinds of planimeters, but they all have certain
fundamental properties in common. The first planimeter was
invented by the Bavarian engineer, J. M. Hermann, in 1814.
Amsler's polar planimeter, which was invented by Jacob Amsler
when a student at Konigsberg in 1854, is the most popular on
account of its simplicity and handiness in use. Thousands of
them have been made at his works in Schaffhausen.
The Amsler planimeter is shown in Fig. 55. It consists of two
bars, (a)the radius bar, and (p)the pole arm, jointedt the point
C The tracing point P, which now coincides with the point B
of thefigure ABDE, is
carried roundthe curve,
and the rollerm,
which partly rolls and partly slips, gives the area of the figure ;
and by means of the graduated dial h, and the vernier v in con-nection
with the roller m, the result is given correctly in four
figures. The sleeve H can be placed in different positions along
the pole-arm h, and fixed by a screw s so as to give readings in
differentrequired units.
A
weightat w is
placed uponthe bar to
* See Note E.
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87-88.] APPROXIMATE INTEGRATION 189
keep the needle point in its place, but in instruments by some
other makers Tis a pivot in a much larger weight, which rests
on the paper. The accuracy of the reading depends upon the
accuracy with which the tracing point follows the curve.*
Fig. 65.
Professor O. Henrici's Report on Planimeters (Report of the British
Association for the Advancement of Science, 1894, pp. 490-523) contains
a sketch of the history of planimeters, the geometrical theory of gener-ting
areas, descriptions of early planimeters, a discussion on Amsler's
planimeter, and a description of some recent planimeters. Professor H. S.
Hele Shaw's paper on Mechanical Integrators (Proceedingsf the Institution
of Civil Engineers, Vol. 82, 1885, pp. 75-143) gives an account of the theory
and the practical advantages of several varieties of planimeters. The descrip-ion
given above is from this paper. An explanation of the theory of
Amsler's planimeter is given by Mr. J. MacFarlane Gray in Carr's Synopsis
of Mathematics. There is a discussion on planimeters in Professor R. C.
Carpenter's Text-bookof Experimental Engineering^ pp. 24-49.
* For the fundamental theory of the planimeter, see Note F, Appendix.
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CHAPTER XII
INTEGRAL CURVES
89.Introduction. A firstintegral
curve was defined in Art. 15.
The student is advised to review that article thoroughly before
proceeding further. In this chapter the subjectfintegral curves
will be studied more fully, and some of their applications to
mechanics will be pointed out. Differentiation under the sign of
integration is an important topic in the integral calculus. Only
a
very specialcase, however, is
necessary
in
what
follows : this
case is considered in Art. 90. Arts. 92, 93, 94, contain an
exposition of the simpler properties of integral curves and a few
examples of their usefulness. Their applications are of especial
value to the student of engineering. For the proper understand-ng
of several of them, a better acquaintance with the theorems
of mechanics is required than some readers of the calculus may
be presumed to have at this stage. Accordingly, a further expo-ition
of the service that may be rendered by these curves is
given in the Appendix for purposes of future reference. Articles
94, 95, discuss the practical plotting of integral curves.*
90. Special case of differentiation under the sign of integration.
A special case of differentiation under the sign of integration
* Arts. 91-96 and the related matter in the Appendix are taken with some
slight but no essential change, from an article entitled Integral Curves,
by Professor W. F. Durand, Principal of the Graduate School of Marine
Engineering and Naval Architecture, Cornell University. The article, which
appeared in the Sibley Journal ofEngineering, January, 1897, is practi-ally
all reproduced here. This chapter has also had the benefit of Professor
Darand's revision.
190
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192 INTEGRAL CALCULUS [Ch.XII.
Therefore, letting A6 approach zero,
^=nj\h-xr-^f(x)d^ (2)
This result will be required in Art. 93.
91. Integral curves defined. Their analytical relations. A more
general definition of an integral curve than that given in Art. 15
willnow be introduced. In
what
follows, a
number ofcurves
will be spoken of together. In order to distinguish between
them, the system of ordinates, that is, the j/'s,for each of the
several curves will be denoted by a subscript number.
If. y=f(s^),
or, for the sake of distinction,
y" =/("') (1)
be the equation of a given curve, the curve whose equation is
y'=\i'y'^''(^"
is called z,firstintegral curve of thecurve
whose equation is (1).The latter is called the fundamental curve. Since I ydx \^ oi
Jo
the second dimension, and yi should be linear, the constant factor
-is introduced in (2),n which a is a linear quantity and has a
magnitude that will make equation (2)convenient for plotting.
It may be called a scale factor. In the definition in Art. 15 the
scale factor was unity.
From (2)on differentiation,
dx a
Hence, as x varies, the slopes of the firstintegral curve vary as
the ordinates of the fundamental ; and therefore the former can
be represented by the latter,and vice versa.
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90-91.] INTEGRAL CURVES 193
The firstintegral curve (2)also has a firstintegral, the latter
has a firstintegral, and so on. These successive integral curves
are called the second, third, etc.,integral curves of the original or
fundamental curve. On using the constant linear quantities, h,
c, " " " w, as scale factors for the sake of plotting the curves con-venientl
and on distinguishing by different subscripts the ordi-
nates that belong to the various curves, the latter will have the
following equations:
Fundamental, y^ =/("); (1)
1 Cfirstintegral, yi "
-\y(i^] (2)
second integral, 2/2 T I yi^" =-r I I ^oC^; (3)
oJo aoJi) Jo
1 /"" 1 /"x /*x r*x
third integral, 2/3= - I 2/2^^ =
^- ( ( \ y^^\ (4)c*/o aoc*/o
*/o "/o
nth integral curve,
yn = - f Vn-idx = " \ ) ) ... fyocbf". (5)wJo aoc"''wJo Jo Jo Jo
From equation (2)
and hence, ^=
^^^ (^)
And in general.
Equation (8)shows that as x varies, the wth derivatives of the
nth integral curve vary as the ordinates of the fundamental
curve ; and therefore, the former can be represented by the latter.
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194 INTEGRAL CALCULUS [Ch. XII.
92. Simple geometrical relations of integral cnryes. In Fig. b^
RP, OA, OBj OCy represent the fundamental, and the first,
second, and third integral curves respectively, whose equations
are (1),(2),(3),(4),f Art. 91.
(a)As X increases, and so long as the fundamental curve RP
lies above the oj-axis,the ordinates of the first integral OA will
increase,
andthe tangent to OA
will makea
positive angle with
"
the aj-axis; when RP lies below the avaxis the tangent to OA
makes a negative angle with the oj-axis; when RP crosses the
a?-axis,the tangent to OA is parallel to the ar-axis. These prop-rties
follow from equations (2)and (6),rt. 91.
(b)At points for which the ordinate of the firstintegral curve
is a maximum or a minimum,-
and there also, by (6),
dx'
yo = 0.
Hence, to a zero value of the ordinate of the fundamental there
corresponds a maximum or a minimum value of an ordinate of
the firstintegral curve.
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92-93.] INTEGRAL CURVES 195
(c)At points where an ordinate of the fundamental is a maxi-um
or a minimum,
and at points where the first integral curve has a point of in-lexion
Differentiation of the members of equation (6)shows that
g =
Owhenf=0.ar.
ax
Hence, to a point on the fundamental at which there is a
maximum or a minimum value of the ordinate, there corresponds
a
point of
inflexion on the firstintegral curve.
93. Simple mechanical relations and applications of integral
curves. Successive moments of an area about ailine. If each in-inites
portion of a plane area be multiplied by its distance
from a given line, the sum of all these products is called the
^momentoffirstegree of the area about the line. If each of the
infinitesimal portions of the area be multiplied by the square of
its distance from the given line, the sum of all the products is
called the moment of second degree of the area with respect to the
line. The latter is the moment of inertia of the area about
the line,examples of which were shown in Art. 80. The moment
of firstdegree is usually called the statical moment. In general,
if each infinitesimal portion of an area be multiplied by the nth
power of its distance from a given line, the sum of all these
products is called the moment of the nth degree of the area about
the line. For the sake of brevity, this may be called the nth
moment.
Thus (Fig.bQ),lay off OX = x^, and erect the ordinate AP at
X, and consider I ^y^dXj the area ORPX, between the funda-
INTEGRAL CALC. 14
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196 INTEGRAL CALCULUS [Ch. Xll.
mental, the axes, and the ordinate AP, If the successive moments
of this area be taken about the ordinate AP for which x = x^ and
these moments be denoted by M^ M^t """, Jlf",in order, then
the firstmoment, ^i = | ^(xi-'X)ydx\ (1)
\ {xi-'X)h/dx; (2)0
the nth moment,-^n
= j *(^i^Tv
^* (3)
In this notation, J^o = ) *(*i^Tv4^f
= j V ^^9 ^^^ area. (4)Jo
(a)Differentiation of (3)with respect to Xi will give by equa-ion
(2),rt. 90,
^=nr\x,-xy-'yd^;dxi Jo
that is, ^=nM^_i. (5)dxi
Hence, M^^^nC^
M^_idxi.
Since dxi is an infinitesimal distance along the aj-axis,it can be
written dx, and hence
M, =
n"'M,_^d^.(6)
By successive application of (6)there will finallybe obtained,
M^^nl P f'-r^Modaf^. (7)Jo Jo Jo
That is,the nth moment of the area ORPX about an ordinate
distant Xi from the origin is equal to factorial n times the nth
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93.] INTEGRAL CURVES 197
integral of the moment of degree zero for the same area. On
substituting for Mq in (7)its value from (4),here is obtained,
Hence, the value of the wth moment of the area of y =/(")above described about an ordinate distant Xi from the origin, is
equal to factorial n times the ordinate of the (n + l)th integral
curve
atx =
Xi] and, therefore, the nth moment may be repre-entedby this ordinate. On using T^^i to denote the ordinate of
the (wH- l)thintegral curve at a? = x^, this may be expressed by
In particular, the statical moment (1)is represented by the
corresponding ordinate of
the
second
integral curve,
andthe mo^
ment of inertia (2)by twice the corresponding ordinate of the
third integral curve. Thus in Fig. 56,
the area ORPX is represented by AX\
its statical moment about AP is represented by BX\
and its moment of inertia about AP by 2 CX,
Suppose that the scale factors used in plotting the three inte-ral
curves, each from the one of next lower order, are a, 6, c,
respectively, as indicated in equations (2),(3),(4),Art. 91.
Then,
zxQ2iORPX=a'AX',
thestatical
momentof
ORPXabout
AP =
ah
" BX]
the moment of inertia of ORPX about AP= 2 abc " CX:
(b) If G is the center of mass (orcenter of gravity)fORPX,
its distance HX from AP, by Art. 79, is determined thus :
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198 INTEGRAL CALCULUS [Ch.XII.
(c) If k is the radius of gyration of the area OBPX about
AP, then by (Art.80),
, o_
Moment of Inertia about AP_
2 abcy^" 2 ftc
^^
Area ayi AX
For further applications to mechanics, and some general re-marks
on the use of these curves in engineering problems, see
Appendix. The reader is recommended to glance at the latter
remarksnow.
94. Practical determination of an integral curve from its funda-ental
curve. The integraph. Suppose that the equation of the
fundamental curve is y =/(").The ordinates of the firstintegral
curve that correspond to successive values x^^ x^ '"f,^n9 of the
abscissas are
- fydx,fldx,.., -r
ydx,
respectively. These may be determined by the ordinary rules
for integration when the functions under the sign of integration
are integrable. If the latter condition does not hold, recourse
can be had to some of the various methods of mechanical and
approximate integration described in Arts. 85-88. It will be
necessary to do this also, when the fundamental has been plotted
merely from a knowledge of the ordinates that correspond to
particular abscissas, the equation of the curve being unknown.
For example, in Fig. 57, the area of each successive section
between the ordinates of the fundamental may be found with a
planimeter, and the ordinates of the integral curve, which is
shown by the dotted line, may be found by successive additions.
As an instrumental check, it is well from time to time to go
around the entire area between the t/-axisand the ordinate in
question, and compare the result with the total area summed to
that point. Numerical means of integration may also be em-ployed.
The trapezoidal rule and the parabolic rule can be
readily used for finding successive increments of area in the case
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93-94.] INTEGRAL CURVES 199
of the fundamental, and hence for finding successive increments
of the ordinates of the firstintegral curve.
In whatever way the integral curve may be derived from the
fundamental, it is well, after plotting, to compare the two and
note the fulfillment of the simple geometrical relations, (a),(b),
Fm, 67.
(c),f Art. 92. Thus, one should look for a maximum or a mini-um
ordinate in the integral corresponding to every zero ordinate
in the fundamental, and for a point of inflexion in the integral
for each maximum or minimum in the fundamental. The tan-ent
of the integral varies with the ordinate of the fundamental,
and hence, the slope of the integral should increase or decrease
when the ordinate of the fundamental increases or decreases.
These relations may be noted in the curves in "Figs. b%j 57.
The integraph is an instrument that is used for drawing the
first integral curve from its fundamental. The theory of it is
given in the Appendix (Note G). It may be used also for
determining the area between a curve and the a?-axis. For the
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200 INTEGRAL CALCULUS [Ch. XII
integral curve can be drawn with the integraph, and the ordinate
corresponding to the area can be measured. Since the length
of the ordinate represents the area, the latter can be found
immediately on making allowance for the scale-factor.
95. The determination of scales. In order to have the various
curves convenient for plotting, it is usually necessary to employ
different scales for the ordinates. If numerical integration is
used,, the value of the area of the fundamental will be found
directly, and the scale may be correspondingly selected so that
the curve will be kept within the desired limits as to size. If
the planimeter is used, the result will be given in square inches
or other area units, and must be converted into the value desired
by the use of a scale factor. Suppose the fundamental plotted
as follows:
horizontally 1 unit of length = p units of abscissa,
vertically 1 unit of "^length = q units of ordinates.
Then 1 unit of area on the diagram will represent pq units of
the integrated function, and the area found must be multiplied
bythis
factor inorder
toreduce
it to thevalue of
the integral
desired. The scale factors a, ", c, etc., may then be chosen as
before.
* See Note G.
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CHAPTER XIII
ORDINARY DIFFERENTIAL EQUATIONS
96. Differential equation, order, degree. A few differential
equations which frequently appear in practical work will now
be discussed very briefly.*
A differentialequation is an equation that involves differentials
or differential coefficients. Ordinary differential equations are
those which contain only one independent variable. For example,
dy = cosxdx, (1)
,
HWV
da?*
y =
x^+ ^, (4)
are ordinary differential equations.
The order of a differential equation is the order of the highest
derivative that appears in it. The degree of a differentialequa-ion
is the degree of the highest derivative when the equation is
* For fuller explanations tjian are given here, reference may be made to
Introductory Course in Differentialquations, by D. A. Murray. (Long-ans,Green, " Co.)
201
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202 INTEGRAL CALCULUS [Ch. XIII.
free from radicals and fractions. Of the examples above, (1)is
of the firstorder and firstdegree, (2)is of the second order and
firstdegree, (3)is of the second order and second degree, (4)is
of the firstorder and second degree, (5)is of the first order and
firstdegree. Differential equations of a very simple kind have
already been considered.
97. Constants of integration. General and particular solutions.
Derivation of a differential equation. If a
relationbetween the
variables together with the derivatives obtained therefrom satis-ies
a differential equation, the relation is called a solution or
integral of the differential equation. For example,
y = m sin x, (1)
y = n cos a?, (2)
y =' Acqs X-i-
B sinx, (3)
y = c sin (a; a), (4)
in which m, w. A, B, c, a, are arbitrary constants, are all solutions
of the equation
This may be verified by substitution. It will be observed that
(5)does not contain vi, n, A, B, c, or a. The solutions of the
differential equations of the first order, which have appeared in
the former part of this book contain one constant of integration ;
those of the second order contain two constants. Examples have "
been given in Arts. 8, 9, 12, 59, etc.
Solutions (1)and (2)above contain one arbitrary constant, and
solutions (3) and (4) each contain two. The question is sug-ested
: How many arbitrary constants should the most general
solution of a differential equation contain ? The answer can in
part be inferred from a consideration of one of the ways in which
a differential equation may arise, namely, by the elimination of
constants, On comparing (3)and (5)it is seen that (5)must
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96-97.] ORDINARY DIFFERENTIAL EQUATIONS 203
have been derived from (3)by the elimination of the two con-stants
A and B, In order to eliminate two quantities, three
equations are necessary. One of these is given, and the others
can be obtained by successive differentiation. Thus,
y =
-4cos X'{- B sin x,
-2= " J. sin a?
-f-J5 cos ",
dx
-4= --
-4cos X " BsiD.X'f
dor
whence, " ^+ y = 0.
In order to eliminate three constants- from a given equation,
four equations are required. Of these, one is given and the
remaining three can be obtained only by successive differentiar
tion. The third differentiation will introduce a differentialcoeffi-ient
of the third order, which accordingly will appear in the
differentialequation that is formed by the elimination of the three
constants. In general, if an integral relation contains n arbitrary
constants, these constants can be eliminated by means of w + 1
equations. The latter consist of the given equation and n rela-ions
obtained by n successive differentiations. The nth differ-ntiation
introduces a differential coefficient of the ntl?order,
which will accordingly appear in the differential equation that
arises on the elimination of the constants. The solution of an
equation of the nth order cannot contain more than n constants ; .
for if it had n
-f-1, their
elimination would
lead to theequation
of the n-f
1th order.*
The solution that contains a number of arbitrary constants
equal to the order of the equation is called the general solution or
the complete integral. Solutions obtained therefrom by giving
* For a proof that the general solution of a differential equation contains
exactly n arbitrary constants, see Introductory Course in Differentialqua-
tions, Art. 3 and Note C, Appendix.
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204 INTEGRAL CALCULUS [Ch.XIII.
particular values to the constants are called particular solutions.
For example, (3)and (4)are general solutions of (5),nd
y = 2cosajH-3sina:, y = 5 cos a: " sin a:, y = 2sin(a; ir),
are particular solutions.
Ex. 1. Eliminate the arbitrary constants m and c from
(1) y = tiMc + c.
Differentiating twice, (2) ^= m,
dx
C3)g0.
These equations may be.interpreted geometrically. If w, c, are both
arbitrary, (1)is the equation of any straight line ; and, therefore, (3)is the
differentialequation of all straight
lines. Ifcis
arbitrary andm has a definite
value, (1)is the equation of any line that has the slope w, and, accordingly,
(2) is the differential equation of all the straight lines that have the slope m.
Ex. 8. Find the differential equation of all circles of radius r.
The equation of any circle of radius r is
in which a, b, the coordinates of the center, are arbitrary. The elimination
of a and h gives
{"-(l)"}'"S.the equation required.
Ex.8. If y = ^aj2 + j5^ prove thata;^-^0.
dx^ dx
Ex. 4. Eliminate c from
y
= cas + c " c*.
Ex. 6. Form the differentialequation of which c^ + 2 cxeff + C^ = 0 is the
complete integral.
Ex. 6. Eliminate the constants from y = ax-\- hx^.
Ex. 7. Form the differential equation which has y = a cos (mx + 6) for
its complete integral, a and h being arbitrary constants.
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97-99.] ORDINARY DIFFERENTIAL EQUATIONS 205
Section I. Equations op the First Order and the First
Degree.
98. Equations in which the variables are easily separable.
If an equation is in the form
its sohition^ obtainable at once by integration, is
Jf.ix)x
-hfAiy)dy = c.
Some equations can easily be put in this form.
Ex. 1. Solve (1 -
x)dy- (1+ y)dx
= 0.
This may be written,
-^ -^-
= 0.^
1 + y 1-a;
This step is called**separation of the variables."
Integrating, log (1 + y) + log (1 -
x) = Ci,
or, (1+ y)(l-
a;) e"i = c.
Ex.2. Solve ^+ JL=J^ 0.
dx ^l-x^
Ex.8. Solve 3"*tanyda5 + (l-c")sec2y(fy 0.
99. Equations homogeneous in a; and y. If an equation is homo-eneous
in X and y, the substitution
y=:vx
will give a differentialequation in v and x in which the variables
are easily separable.
Ex. 1. Solve (x*+ ^)dx -2xydy= 0,
Rearranging. (1)| = ^On putting y = vx,
dx dx
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206 INTEGRAL CALCULUS [Ch.XIII.
Sabstitution of these values in (1) gives
'
dv 1 + r*
dz 2v
Separating the variables,^
-
^^^= 0.
X 1 " r*
Integrating, log a;(l r^)log c ;
that is,
loga;(l-?!Wlogc;
whence, a^-
y*=
ex.
Ex. S. Solve y^dx + (xy + x^)dy = 0.
Ex. 8. Solvex^dx-{7fi -\-y^)dy
0.
Ex. 4. Show that the non-homogeneous equation of the firstdegree in x
Budy
dy_
ax-\-hy-\-c
dx a'x + b'y + c'
is made homogeneous, and therefore integrable, by the substitution
x = xi + h, y = yi + kj
hf k being solutions of ah -\-bk
-\-c = 0^
a'h + b'k + c' = 0.
100. Exact differentialequations. An exact differential equa-ion
is one that is formed by equating an exact differential to
zero. It follows from Art. 24 that
Mdx-h
Ndy = 0
is an exact differentialequation if
dy dx
Ex. 1. Solve (a2-2xy- y^)dx- (" + y^dy = 0.
Ex. 3. Solve (x^ ixy-2y^)
dx + (y^ 4xy-2x^)dy
= 0.
Ex. 8. Solve (2xhf + iofi-I2xy^
+ Sy^-a^ + e^^dy
+ (l2x^y-\-2xy^-{'43fi-4y^-^2ye^-eif)dx 0.
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99-101.] ORDINARY DIFFERENTIAL EQUATIONS 207
101. Equations made exact by means of integrating factors. The
differentialequation
is not exact. Multiplication by " gives
This is exact, and its solution is- = c, or a? = (^.y
When multiplied by " f the firstequation becomes
xy
X y*
which is also exact. The solution is
logo? " logy = log c,
whence,- = (^ or a? = cy.
Another factor that will make the given equation exact is
--. Any factor such as ","
," employed above, which changes
aj* f ix^y
an equation into an exact differential equation, is called an
integrating factor. It can be shown that the number of integi^at-
ing factors is infinite. There are several rules for finding
integrating factors. In the following examples, the necessary
integrating factorcan be found by inspection.
Ex. 1. Solve y dx " xdy -\-\ogxdx = 0.
Here, logxdx is integrable, but a factor is needed for ydx " xdy.
Obviously " is the factor to be employed, as it will not affect the third term
x^1
injuriouslyrom the point of view of integration. On multiplication by "
the given equation becomes
ydx-xdy . loga;_Q
x2"
as*
*
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208 INTEGRAL CALCULUS [Ch.Xlll.
The solution of this equation reduces to
cx + y " log* " 1 = 0.
Ex. 2. a (xdy + 2 y dx) = xydy.
Ex.8. (x2+ y2+ l)dJB-2a;ydy = 0.
Ex.4. (x"c*-2wy2)dx + 2ma;ydy = 0.
102. Linear equations. If the dependent variable and its de-ivati
appear only in the firstdegree in a differential equation,
the latter is said to be linear. The form of the linear equation
of the firstorder is
|+iV=Q, (1)
in which P and Q are functions of x, or constants. The linear
equation occurs very frequently. The solution of
dx
that is,of^
=
^PdXyy
- \pdat \pd*IS y=:ce ^
,OT ye* =c.
On differentiation the latter form gives
J'^(dy-^Pyd^)0,
which shows that e''^ is an integrating factor of (1).
Multiplication of (1)by this factor gives
J'^idy"{-Pyda)
==J'^Qdx',
and this, on integration, gives
yeJ^^ = jgJ^'Q da?
-I-c,
or y =
e-^'''^{fJ'''^Qdx-^c}.2)
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101-103.] ORDINARY DIFFERENTIAL EQUATIONS 209
The latter can be used as a formula for obtaining the value
of 3/ in a linear equation of the form (1).
Ex. 1. Solvejc^-y =:sc8.
dx
This is linear since y and -^appear only in the firstdegree. On putting
dx
it in the ordinary form (1),it becomes
ax X
1 \pd* " r*^ ^1
Here P = "
, andhence, the integrating factor e = "
*= e'"'" = -.
X X
By using this factor, and adopting the differential form, the equation is
changed into
-dy"
-ydx= xdx,
X x^
Integrating, 2f= ^^ + c, or y =
-x* + ex.
"B2 2
Ex.2. Solve !^+y=:e-".dx
Ex. 8. Solve ^+ ^"^y = 1.
dx "2
Ex.4. Solve
^4--^y=
dx x^ + r (a;a+l)8
Ex.6. Solve ^+
iiy="
dx X xn
103. Equations reducible to the linear form. Sometimes equa-ions
that are not linear can be reduced to the linear form. One
type of such equations is
in which P, Q, are functions of x, and n is any constant. Divi-ion
by y" and multiplicationby (" n H- 1)gives
(- n +l)2/-"g+
- n + l)Pr"^i = (- n-f-1)Q. .
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210 INTEGRAL CALCULUS [Ch. XllL
On substituting v for ^"^^ this reduces to
^ + (l-n)Pv={l-n)Q,
which is linear in v.
Ex.1. Solve ^+iy
= *y.dx X
Division by y* gives y*^ +-jr*
= a^.dx X
On putting v for y^^ this takes the linear form
dv 5^ft^2
V = " OX*.
dx X
The solution is v = y* = ex* + fo^.
Ex. S. Solve ^+?y
= 3a;*y*dx X
Ex.3. Solve ^+
-^L= xyi
Ex. 4. Solve 3^ +-?-y
=^
daj 05 + 1 y*
Ex. 6. Show that the equation
in which P, Q, are functions of x, becomes linear on the substitution of v
Section II. Equations op the First Order but not of
THE First Degree.
104. Equations that can be resolved into component equations of
the first degree. In what follows,-^
will be denoted by p. Thedx
type of the equation of the firstorder and nth degree is
which on expansion becomes
p- + P,p-' + P^"-2 + ... + P".ip + P^ = 0. (1)
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103-105.] ORDINARY DIFFERENTIAL EQUATIONS 211
Suppose that the first member of (1) can be resolved into
rational factors of the firstdegree so that (1)takes the form,
(i"-i?0(l"-^2)-(i"-i?")= 0. (2)
Equation (1)is satisfied by any values of y that will make a
factor of the first member of (2)equal to zero. Therefore, in
order to obtain the solutions of (1),equate each of the factors
in (2)to zero, and find the integrals of the n equations thus
formed. Suppose that the solutions derived from (2)are
fi(x,y,(h)0, /2(aJ,, C2)=0, ..., /"(a?,2/,c")=0,
in which Cj, Cg, """, c^ are arbitrary constants of integration. These
solutions are justas general if Cj = C2 = """ = c^ since each of the
c's can take any one of an infinite number of values. The solu-ions
will then be written
/i(a?,f c)= 0, /2(a?,, c)= 0, ..., /"(a?,, c) 0,
or simply, /i(a?,, c)f^a?,, c)"
"/"(a?,, c)
0.
Ex. 1. Solve f V+ (X 4- y)^ + "y = 0.
\dxl ax
This equation can be written (P + y)(l"+ x)= 0.
The component equations are l" + y = 0, p + x = 0,
of which the solutions are log y + " + c = 0, 2 y + x* + 2 c = 0.
The combined solution is (logy 4- " + c)(2y 4- a;^ + 2c)= 0.
Ex. 3. Solve (^ ) = ax*. Ex. 8. Solve p8 + 2 xp2 _ ^2^2 _ 2xy^ = 0.
105. Equations solvable for y. When equation (1),Art. 104,
cannot be resolved into component equations, it may be solvable
for y. In this case,
can be put in the form y = F(x,p).
Differentiation with respect to x gives
P
nTTBGRAL CALC. " 15
=
*('^'^'i)
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212 INTEGRAL CALCULUS [Ch. XIII.
which is an equation in two variables x and p. From this it may
be possible to deduce a relation
^(a?,p,)=0.
The elimination of p between the latter and the original equa-ion
gives a relation that involves a?, y, c. This is the solution
required. If the elimination of p is not easily practicable, the
values of x and y in terms of 2" as a parameter can be found, and
these together will constitute the solution.
Ex. 1. Solve x " ffp = ap'^.
Here y =^"
P
Differentiating with respect to x, and clearing of fractions,
(ap"+
x)J=D(l-l)").Tliis can be pat in the linear form
dx 1_
ap
Solving, X =^
{c+ a sin-^p).
Substituting in the value of y above,
y = -
ap-\-"=(6 + a sin-ip),
Ex. 3. Solve 4y = jB"+ pa.
Ex. 8. Solve y = 2p + 3p2.
106. Equations
solvable
for x. In this case
f(x,y, p)can be
put in the form
x=zF(rfyp).
Differentiation with respect to y gives
r"-l"from which a relation between p and y may sometimes be obtained,
say, f(yyP,c)=0,
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105-107.] ORDINARY DIFFERENTIAL EQUATIONS 213
Between this and the given equation p may be eliminated, or x
and y may be expressed in terms oip as in the last article.
Ex. 1. Solve x = y + a logp.
Ex.3. Solve x(l+p*)=l.
107. Clairaut's equation. Any differential equation of the first
order which is in the first degree in x and y comes under the
cases discussed in Arts. 105, 106. An important equation of this
kind is thatof
Clairaut.* It has the form
y=px+f{p). (1)
Differentiation with respect to x gives
From this.p^p+\x+np)\f^.
ar+/'(p)
0, (2)
1 = 0. (3)
From the latter equation it follows that p = c. Substitution of
this value in the given equation shows that
y =
cx+f(c) (4)
is the general solution. See Introductory course in differential
equations, Art. 28, for remark on equation (2). Some equations
are reducible to Clairaut's form, for instance, Ex. 2 below.
Solution (4)represents a family of straight lines. The en-velope
of this family of lines will also satisfy the differential
equation, since a;, y, p, at any point on the envelope is identical
with the ", y, p of some point on one of the tangent lines of which
(4)is the equation. The equation of the envelope of (4)is called
the singular solution of (1). Singular solutions are discussed in
Chapter IV. of the work referred to above.
* Alexis Claude Clairaut (1713-1766) was the first person who had the
idea of aiding the integration of differential equations by differentiating
them. He applied it to the equation that now bears his name, and published
the method in 1734.
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214 INTEGRAL CALCULUS [Ch. Xlll.
Ex. 1. Solve y-px + aVl+jp^.
The general solation, obtained by the substitution of c for p, is
y=:cx + aVl + c^
which represents a family of lines. The envelope of this family is the circle
a;2 + ys = a^.
The latter equation is the singular solution.
Ex. 8. x\y-px)=t/jif^.
On putting x^ = u, y^ = ", the equation becomes
which is Clairaut^s form. The solution is
that is, y^ = ca^ + "^.
Ex. 8. y =pz + sin-ip.
Ex. 4. py =p2x + TO.
Ex. 6. xy^ =pyx^ + x H-py.
108. Geometrical applications. Orthogonal trajectories. curve
is often defined by some property whose expression takes the
form of a differential equation. In the examples given below the
differential equations of the curves are of a less simple character
than those which appeared in similar problems in Arts. 8, 12, 32.
Problems that relate to orthogonal trajectoriesre of consider-ble
importance. Suppose that there is a singly infinite system
of curves
/(a?,2^,a)=0, (1)
in which a is a variable parameter. The curves which cut all the
curves of the given system at right angles are called orthogonal
trajectoriesf the system. The elimination of a from (1)gives
an equation of the form
"(",|)=0, (.)
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107-108.] ORDINAHY DIFFERENTIAL EQUATIONS 216
the differential equation of the given family of curves. If two
curves cut at right angles, and if 0i,"^2"the angles which the
tangents at the intersection make with the axis of Xy then
and therefore, tan 4"i= " cot "^
Hence,
-^
for one curve is the same as for the other.
dx dy
dx
Therefore, the differential equation of the system of orthogonal
trajectoriess obtained by substituting
" " f ^dy dx
in equation (2). This gives
4"(",-!)=".Integration will give the equation in the ordinary form.
Suppose that /(r, ,c)
= 0
is the equation of the given family in polar coordinates, and that
is the corresponding differential equation obtained by the eliminar
tion of the arbitrary constant c. Leti/^i,p^
denote the angles
which the tangents to one of the original curves and a trajectory
at their point of intersection make with the radius vector to the
point. Then
tani/fj
" cot \ff^
dBNow tan^i = r" . Hence the differential equation of the
dr
required family of trajectoriess obtained by substituting
r dO 'dr
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216 lifTEGItAL CALCULUS [Ch.Xllt
that is,
-r^^for^dr dO
in (3). This gives 4/r,,-
r^j^as the differential equation of the orthogonal system.
Ex. 1. Find the orthogonal system of the family of parabolas
Differentiating,y^
= 2 a,
dx
and eliminating a, y = 2x^"dx
This is the differentialequation of the given family. Substitation of
dy dx
ndx
gives y =
-2a;" ,
the differential equation of the family of trajectories.ntegration gives
the equation of a family of ellipses whose foci are on the y-axis, and whose
centers are at the origin.
Ex. 2. Find the orthogonal trajectoriesf the cardioids
r = o(l"
COBB).
Differentiating, " = a sin ^.
dO
dv (t
Elimination of a gives -i.= r cot ^"
dB 2
the differential equation of the given family of curves. Therefore, the equa-ion
of the system of trajectoriess
-r^= cot?.
dr 2
Integration gives r = c (1 + cos ^),
another system of cardioids.
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108.] ORDINARY DIFFERENTIAL EQUATIONS 217
Ex. 8. Find the curve in which the perpendicular upon a tangent from
the foot of the ordinate of the point of contact is constant and equal to a,
determining the constant of integration in such a manner that the curve shall
cut the axis of y at right angles.
Ex. 4. Find the curve whose tangents cut off intercepts from the axes the
sum of which is constant
Ex. 6. Find the curve in which the perpendicular from the origin upon
any tangent is of constant length a.
Ex. 6. Find the curve in which the perpendicular from the origin upon
the tangent is equal to the abscissa of the point of contact.
Ex. 7. Find the orthogonal trajectoriesf the straight lines y = cx,
Ex. 8. Find the curves orthogonal to the circles that touch the y-axis at
the origin.
Ex. 9. Find the orthogonal trajectoriesf the family of hyperbolas
zy = A;2.
Ex. 10. Find the orthogonal trajectoriesf the ellipses
in which X is arbitrary.
^t y
-1
Ex. 11. Show that the system of confocal and coaxial parabolas
y*= 4
a(x+
a)is
self-orthogonal.
Ex. 12. Find the orthogonal trajectoriesf the system of circles
y = c cos d,
which pass through the origin and have their centers on the initialline.
Ex. 18. Find the orthogonal trajectoriesf the system of curves
y^sin nd = a".
Ex. 14. Find the equation of the system of orthogonal trajectoriesf the
2 afamily of confocal and coaxial parabolas r = " "
1+ cos ^
Ex. 15. Determine the orthogonal trajectoriesf the system of curves
)"" = a" cos n9 ; and therefrom find the orthogonal trajectoriesf the series of
lemniscates r^ = a^ cos 2 $,
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"3 ^fx -Tit g-rnnaiBif rf "ria-
"", 4 ^4y* Z/"*^^Pra-";=0, sobjeettoAe oonditkn that f = 0
110" Sqiuiti4MMof tlieform ^^ =/(y)*Multiplication of both
\\\m\\wiv%ot th\H equation by 2^^-ives(IX
rntfl"mtinK, (dxj^^f-^^^'^'^'^'^
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220 INTEGRAL CALCULUS [Ch.XIU.
an equation of the firstorder between p and x. Its solutiongives
p in terms of x ; thus,
The value of y can be found from this by successive integration.
Ex. 1. Find the curve whose radius of curvature is constant and equal
to J?.
The expression of the given condition gives
{" (!)"}"
Substituting p for ^, clearing of fractions,and separating the variables,dx
dp dx
4 ^
Integrating, ^= "
x " a
in which a is an arbitrary constant of integration.
Solvingfori), t, = ^ = ^^-"*
^ Vi22-(x-a)"
whence, y-b = " Vija-(x- a)3,
in which b is the arbitrary constant of integration. This result may be
written
(a;-rt)2(y-6)2 = iP.
The integralrepresents allcirclesof radius B,
Ex.2. Solve ^-o (i)"="-
Ex.8. Solve ^^ = 2.
dx^ dx^
112. Equations of the second order with one variable absent.
(a)Equations of the form
i^-%')""- w
s
\
\
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111-112.] ORDINAHY DIFFERENTIAL EQUATIONS 221
on the substitution of p for
-^,become
dx'
/(|,i",^)0, (2)
an equation of the firstorder in p and x. Suppose that the solu-ion
of (2)is
Then the solution of (1)is y = jF(xy Ci)a?+ c^
(6)Equations of the form
ff^, y\=0, (3)
on thesubstitution of p
for
-^,become
dx
/(l"|,P,.)=0,an equation of the firstorder in p and y. Suppose that its solu-ion
is
Then the solution of (3)is
JF(y,c)^*
Equations of the type in Art. 110 and Ex. 1, Art. Ill, are ex-amples
of the equations discussed in this article.
Ex.1. Solvea;^ +^
= 0.dx^ dx
Ex. 8. Solve ^+a"^ = 0.
dx^ dx^
Ex. 8.. Solvey^ + f^y=l.
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222 INTEGRAL CALCULUS [Ch. XIII.
113. Linear equations. General properties. Complementary Func-ion.
Particular Integral.
The form of the linear equation of order n is
g+i'.^^i".^^-+i-^.|+i-^-1).
where Pi, Pj,---, P,^ X, are constants or functions of x. The
linear equation of the firstorder was considered in Art 102,
The completeintegral
of
is contained in the complete solution of (1). Ify=fi(x)
be an
integral of (2),hen, as may be seen on substitution, y =
Ci/i(aj)Ci being an arbitrary constant, is also an integral. Similarly,
ify=M^)y y=M^)j """" y=fn(^)"
be integrals of (2),then
y =
Cif2{^)"""" y="^n/"(")"herein c^, """, c^ are arbitrary con-stants,
are integrals of (2). Moreover, substitution will show
that
y =
"hMx) + c^/aCaj)... + cj;(aj) (3)
is an integral. If
/i("),/2("),""""/"a?),re linearly independent,
(3)is the complete integral of (2),ince it contains n arbitrary
constants.
Ify==F{x)
be a solution of (1),hen y = Y-\-F(x), (4)
in which F=c,/,(x)+ (^^{x) h c^fn(x),
is also a solution of (1). For, the substitution of Y for y in the
first member of (1)gives zero, and that of F{x) for y, by hy-othesis
gives X, Since the solution (4)contains n arbitrary con-stants,
it is the complete solution of (1). The part Y is called
the complementary function,and the part F{x) is called the par-icular
integral. Equations of theform
(2)willbe
considered in
the articles that follow.
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113-114.] ORDINARY DIFFERENTIAL EQUATIONS 223
114. The linear equation with constant coefficients and second
member zero. On the substitution of e"^ for y^ the first mem-ber
of
becomes (m*+ Pim**"^ H h P^-iWi + Pn)e^.
This expression is equal to zero if
mr + PimT"' + .." + P.^im -h ^= 0. (2)
The latter may be called the auxiliary eqaation. Therefore, if
mi be a root of (2), = e"*!* is an integral of (1); and if the n
roots of (2)be m^ m^ """, m^ the complete solution of (1)is
y = Cie-H* + Cje-v H 1-c"e"" (3)
Ex.1. Solve ^-2f?-86y 0.
The auxiliary equation is m'^ " 2 w " 36 = 0 ;
and itsroots are m = " 5, m = 7.
Hence, the complete solution is
y = cie-*" + Cjc^*.
If the auxiliary equation has a pair of imaginary roots, say
mi = a + t)3,m2 = a "
i)8 (idenoting V" 1),the corresponding
part of the integral can be put in a real form. For,
=
e*'|ci(cos)3aj1
sin fix)-f C2(cos)8a?isin^Sa?)}= e"* {A cos px-\-B sin )8a:).
Ex.8.
Solve^+8^+26y0.
The auxiliary equation is
w2 + 8 w + 26 =s 0,
and its roots are m=" 4 + 3i, m=:" 4 " 3i.
Hence, the integral is y = c-4" (cicosSx + C28in 3a;).
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224 INTEGRAL CALCULUS [Ch. XIII.
Ex. 8. Solve2^ 6^-12" = 0.
Ex. 4. Solve 0 + 8y = 0.
Ex.5.Solved-^-
6y = 0.
E.,.Sol.eg-3S-6|8, = 0.
Ex.7. Solve
^-a*y0.
115. Case of the auxiliary equation having equal roots. If two
roots of (2)Art. 114 are equal, say mi and mj, solution (3)becomes
y = (ci-j-cs)"""' + Cge"^ H h c^e^.
Since Ci + c^ is equivalent to a single constant, this solution has
n " 1 arbitrary constants, and hence, it is not the general solu-ion.
In order to obtain the complete solution in this case, make
the substitution
wij = Wi-|-
h.
The terms of the solution that correspond to mi, m^ will then be
y = Cie"^* H- C2e^'"i+*^*,
that is,. y = "r^' (ci+ Cje**).
.
On expanding e** in the exponential series, this becomes
=
e"*i'(j-J5aj+ 1 02^W+ terms in ascending powers of h),
in which-4
=Ci H- Cg, and J5 =
c^,
Now let A approach zero, and solution (3),rt. 114, takes the
form
y =
e"'i'(-4Bx) H- cjge"-* H h c^c"^;
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114-116.] ORDINARY DIFFERENTIAL EQUATIONS 225
for the arbitrary constants c" C2 can be chosen so that Ay B will
be finite,and that c^^,etc., will approach zero. If the auxiliary
equation have three roots equal to mi, it can be shown that the
corresponding solution is
y =
e"H*(ci-fjjaj-f-Csa^,
and, if it have r equal roots, that the corresponding solution is
y = e"i* (ci-f-Cjic H h c^_iar-^).
If a
pair ofimaginary
roots,a
+ ip,at, "
i)3,occurs
twice, the
corresponding solution is
y = (ci Cja?)e"*+V""(c + C4a;)e"*-'^^",
which reduces to
y =
e^\{A + A^x) cos fix+{B + BjX)sin^Sa?}.
Ex.l. Solve ^-3^+ 3^-y = 0.
(to* dx'^ dx
The auxiliary equation is
m"-3m2H-3m-l=0,
of which the roots are + 1 repeated three times. Hence, the solution is
y = e"(ci+ Cax + CsflB^).
Ex.8. Solve ^+ 2^+*? = 0.
dofi dx^ dx
Ex.8. Solve ^~^-9^-ll^-4y = 0.dx^ dx^ dx^ dx
116. The homogeneous linear equation with the second member
zero. This equation has the form
wherein Pi, j"2,""-,/"""^re constants.
(a)First method of solution. If the substitution
z = log X, that is, a? = e*,
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226 INTEGRAL CALCULUS [Ch.XIU.
be made, equation (1)will be transformed into
wherein qi, q^, ""-, g^, are constants. This can be easily verified.
Ex.1. Solve
x2^-a;^+y= 0.
dx^ dx
If " = logx, then ? =-, and
dx z
d}[_dji dz
__ldydx dz dx xdz
dx^ dz\dx)dxst^i\dx dz)'
Substitution of these values m the given equation changes itinto
the solution of which is y = e*(ci+ C2"),
whence, y = x(ci+ cj logx).
(b) Second method of solution. The substitution of oT for y in
the firstmember of (1)gives
{m(m " l)-"(m " n + l)+pim(m -- !)"""(m "
"4-2) + """
This is zero, and accordingly y = af' is a solution of (1),f
m(m" l)--.(m n-^ l)-j-j"im(m T)"- (m^n'\-2)-\
+ i"n-im+i"n = 0. (3)
Hence, if m^ mj, """, m^, are the roots of (3),he complete solu-ion
of (1)is
y = CiQCT^4- c^ H h C^n,
To a solution y =
e""(ci CjS: + """ H- c,._iaf-^)f (2),here cor-responds
a solution.v
=
aj*(cih CglogajH- """ +c^-i(loga:)'"-^)f
(1),since 2 = log a?. It can be easily shown that the auxiliary
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116.] ORDINARY DIFFERENTIAL EQUATIONS 227
equation of (2)is identical with (3). Hence, if m^ is repeated r
times as a root of (3),he corresponding solution of (1)is
y =
oj^i^CiCj log a; -f...
+ c^_i(logj)'"^}.
Ex.2. Solvea;8^+3a;2^ +
^+ y = 0.
Substitution of x* for y gives (m^ -f l)x""0,
of which the roots are-1,
1":^ Lll2^.2
'
2
Hencethe solution
is
2/-=^+ x*{2
cosf^ogx\+
C8
sin/:^ogx\|
Ex. 3.x2^+4x^
+ 2y = 0.dx^ dx
.
Ex.4.x2^-3x*^4-4y
= 0.dx^ dx
Ex.6.
x3^-3x2^+7x^-8y= 0.
dx" dx^ dx^
EXAMPLES
1. If y = Ae^ + Be-'", prove that^-m
= 0.
2. Derive the differentialequation of all circles which pass through the
origin and whose centers are on the x-axis.
3. sec2 X tan ydx-^ sec^ y tan x dy = 0.
4. xdx + ydy = a^''^y-y^.x2 + y2
5 (?x_
dy
x^-2xy~y^-2xy
6. (2ax + by-hg)dx-h(2cy-hhx-{-e)dy = 0:
7. (l-{-xy)ydx-h(l-xy)xdy
= 0,
.8.
y(2 xy + C) (?x - 6'(?y = 0.
9.
.!-",. + 1.
10. cos2 x^^ + y = tan X.
dx
INTEGRAL CALC. " 16
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228 INTEGRAL CALCULUS [Ch. Xlll.
11.
X(l-x2)^+(2x2_l)y=:aX".12.
3^+ yco8x = y"8in2x.
13. ^ = x"y8-xy.
14.p8(x+ 2y)H-3i)2(x+ y)+ (y+ 2x)p = 0.
15. xp2-2yp + ax = 0.
16. p2y + 2px = y. ^4.t/-0dx8
^~
17. 6"'(p-l) + p"e"i'= 0.
(?X2
M. ^-4^1+ 8^,-8^4^=0.
dx* dx" dx* dy^
85. ^-2^ + ^ = 0.dx* dx" dx*
"" fa-")-^'S="- 29.EI^
=
-ybx,dx*
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APPENDIX
NOTE A
[Thisnote is supplementary to Art. 33]
A method of decomposing a rational fraction into its partial
fractions.
Suppose that ^. (/^
"
r- is a proper rational fraction. The
substitution of a for x in this fraction, {x^af being left un-changed
and the subtraction of the fraction thus formed gives
m m_Ax)F(a)^f(a)F(x)F(x)(x-ay F(a)(x-ay'~ F(a)F(x){x-ay
'
^^
Thenumerator of
the fraction in thesecond member of (1)
vanishes for x^a, and hence it is divisible by a? " a. Let the
quotient be denoted by"l"(x).
hen
f(x)_.,
f(a) 1 i^jx)
F{x)(x-ay"
F(a)(x -
a)""^
F{d)'
F{x)(x-
a)""^'^^
Of the two fractions in the second member of (2)the firstis
one of the partial fractions required, and the second has a de-ominat
of lower degree than the original fraction possesses.
The second fraction can be similarly decomposed, and by the
repetition of the operation all the partial fractions will be
found.
When the factors of the denominator are all different and
of the first degree, the ^decomposition of a fraction can be
effected very quickly. For example, on taking the fraction
229
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230 INTEGRAL CALCULUS [Note A.
7 ./
^
\.-, r^ and substituting a for x except in x " a,
(x "
a)(x" b)(x c)
^ ^
and subtracting the fraction thus formed, there is obtained
/(^) /(g)^
F(x)
(x'"a)(x^
b)(x "
c) (a? a)(a" 6)(a "
c) ("" ft){x"cf
in which F(x) is a constant or of the first degree in x.
The partial fraction whose denominator is x " a, which is
formed by thisrule,
isaccordingly ^1^21
The
partial fractions whose denominators are (x " b),(x"
c),canbe
written by symmetry. This is easily verified. For, on assuming
(x"
a){x" b){x"
c) x " d x " b x " c
and clearing offractions,
f(x)= ^ (a? 6)("-
c)-I-J5(a? a)(a? c)H-
C (a? a)(a? 6).
The substitution of a for x gives
/(a)=
-4(a" 6)(a"
c)j
whence,
A =
/(")
(a" b)(a c)
It will be found, on putting 6 for x that
B= m,
(b"a)(b-c)'
and on patting c for iK, that
c= M
(c d)(c b)
Therefore,
(a? d)(x b)(x c) (x"
a)(a b)(a c)
_
.^m+
m.+ 7^
.;""'..
"
-.+(b-a)(x-b)ib-c) (c-a)(c-b)(x-c)
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Note A-B.] APPENDIX 231
Ex. 1.
2a;g-l_
2.22-1 2(-3)"--l
(a; 2)(a; 3)(x- 6) {x-
2)(2 +3)(2-
6) (-3 -2)(3 + a;)(-3 -6)
'^(6-2)(6 + 3)(x-6)
7^
17^
49
16(a; 2) 40 (aj+ 3) 24 (a; 6)
jj 23a? + 2 3-2 + 2
_"'
'
(z-2)2(x-3) (X- 2)2(2-3)
On determining F by subtraction,
3x + 2 8^
11
(X - 2)2(x - 3)"^
(X - 2)2 (X - 2)(X - 3)
11 11
(3-2)(x-3) (x-2)(2-3)
Hence,_l"+2_^
8_
_1]L-_1L_.(x-2)2(x-3) (a-2)2
X-3 x-2
NOTE B
[Thisnote is supplementary to Art. 46]
To find reduction formula for jx*^(a + bx*^)Pdx by integra-ion
by parts. In what follows jx'^(a baf^ydxwill be denoted
by /.
(a)On
putting
dv =
Qif^\a-
boi^yx, u =
af*""+\
it follows that v = %t-^^^\ du = (m - n-i-
1)af-" dx,
nb (p-^1)
Hence, 7=
,\
^
^/". "
-^I af *(a+ 6aj")^^da?.
But aJ"-*(a 6aj")'+iar-"(a 6af")(a6af")'
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234 INTEGRAL CALCULUS [Notb C-D.
From this, on solving for /,
sin^-^ajcos^+^oj ,m " 1 C^-m-t,
Join
m; UUS "/,
7f* J. I
^;_m-S
^ ^^^n ^J^
=
H I Sin"*ajcos*a?aa5.
f^-1
From this result, on transposition and division by "
,
fsin"*-^cos" xdx =
^^^^^"^ ^
^^f'^'^+ ^^^^^ rsin"? cos" a: da:;J m " 1 m " 1"/
whence, on changing m into m-\-2j
*/m + 1 mH-l"/
Formula C, Art. 51, can be obtained by writing
/= Icos'*"^ajsin*ajd(sinaj),
putting dv = sin"*x d (sina?), = cos"~^ a?,
and then integrating by parts and reducing.
Formula Z",Art. 51, can be derived from C by transposition and
the change of n into n-\-2.
NOTE D
[This note is supplementary to Art. 67]
It is explained in the differential calculus that if the differ-nce
between two quantities be infinitesimal compared with
either ofthem, then the limit
oftheir
ratio
is
unity, and either
of them can be replaced by the other in any expression involving
the quantities. A deduction that can be made by means of this
principle is of great importance in the practical applications of
the integral calculus.
Ifai-f-a2H ha"
represent the sum of a number of infinitesimal quantities which
approaches a finite limit as n is increased indefinitely.
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notbD-b.] appendix 236
and if jSi,p^....,
^"
be another system of infinitesimal quantities, such that
where ei, eg, """, e"
are infinitesimal quantities, then the limit of the sum of Pu p^t
""', Pn is equal to the limit of the sum of "!, Oj, """, a".
It follows from equations (1)that
A+i82+ - +Pn=("^i+^+ - +a")+ (aiei+a2e2+ +anen).(2)
Letrj
be one of the infinitesimal quantities Ci, 62, """, e,, which
is not less than any one of the others. Then
Wl+
P2+- +
Pn) ("!+ 02 + - +
"")"("!+ "2 + - +
"n)1/-
But by hypothesis "! + 02 H- """ + "n 1^3,8 finitelimit, and hence
the second member of this inequality is infinitesimal. There-ore
the limit of jSi+ ^82+ """i^i.is the same as the limit of
NOTE E
[Thisnote is supplementary to Arts. 84-87]
Further rules for the approximate determination of areas. A
few more rules for approximately determining the area of
LAKT (Fig.63) may be stated. As before, h denotes the
interval between successive equidistant ordinates, and merely
the coefficients of the successive ordinates are given in the
formulae. In the trapezoidal rule, strips were taken in which
two ordinates were drawn; in other words, the ordinates were
taken by twos. In the parabolic rule, strips were taken in
which three ordinates were drawn; that is, the ordinates were
taken by threes.
* See B. Williamson, Treatise on the DifferentialCalculus^ Arts. 38-40.
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Note E-F.] APPENhlX 237
found by integration. For example, in the case of each succes-sive
pair of ordinates in Art. 85, the given arc was replaced by a
straight line, and in the case of each successive group of three
ordinates in Art. 86, the given arc was replaced by the arc of a
parabola. On assuming that the equation of the second curve is
y = A +-4iaj
+ ^2aj'+-+A"", (1)
the coefficientsA^ A^, A^ """, A^, can be determined. For, the
substitutionin
(1)of the coordinates of then
+1
given points,
namely the extremities of the given equidistant ordinates, will
give w + 1 equations, by means of which the values of the n + 1
coefficients A^ A^, """,A^, can be found.* If n is sufficiently
great, the difference between the area of the second curve and
that of the original curve will generally be very small. The
general
formula for the case
ofn + 1
equidistant ordinatescan
also be deduced by the method of finite differences.!For a dis-ussion
on various methods of finding an approximate value of a
definite integral by numerical calculation, reference may be made
to J. Bertrand, Calcvl IntSgraly Chapter XII., pp. 331-352.
NOTE F
[Thisnote is supplementary to Art. 88]
The Fundamental Theory of the PlanimeterJ
In Fig. 58, ALBO is a plane figure whose area is required, and
QX is a given straight line taken as the axis of X Let MK rep-esent
a plate of which two given points always move, Q along
QXy and P on the contour of the given area. Then QP is a
straight line fixed with reference to the instrument. Let b be
the length of QP. Let W be the recording wheel with axis
parallel to QP. Its actual location is arbitrary.
* Also see Lamb, Infinitesimalalculus^ Art. 112.
t See Boole,Calculus
ofFinite
Differences hapter III,Arts.
10-14.
X This note is by Professor W. F. Durand, who has kindly permitted its
insertion here.
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238 INTEGRAL CALCULUS [Note F.
The movement of P from P to G, a point very near, may be
decomposed into: (1)A movement dx parallel to QX, (2)a
movement dy at right angles to QX. It will first be shown
that the record of the wheel W due to the dy component will, for
the closed area, be zero.
Fio. 68.
It may be noted that the amount of the dy record depends on
the dy and on the configuration of the instrument under which it
is traversed. Now it is evident that for every dy traversed in
the up direction, there will be an equal dy traversed in the down
direction, and under the same configuration. In the diagram the
pair thus traversed is dy and dyi. The net record for such a pair
is zero, and for every other pair, zero, and therefore, for the
entire contour, zero.
It follows that the entire record will be merely that due to the
dx components. This is found as follows. The component of
dx in the direction of the plane of the wheel is dx " sin 0, But
sin^ =^. Denote that part of the record diie to dx by dB.b
Then,
dB=dx'sm6
=_ydx
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Note F.] APPENDIX 239
Hence, E =
-(ydx=z",
and therefore A = bB.
It only remains therefore to graduate W conformably to the
length of 6, or, vice versa, to graduate W and give to b an appro-riate
length. The latter is the usual method. By giving to b
various lengths, the area may be read off in corresponding units.
Thus far it has been assumed that Q follows the straight line
QX, It will next be shown that the record is independent of the
path of Q so long as it is back and forth on the same line.
Fig.
To this end let FM (Fig.59) be any area, and ABODE a
broken line. I^et A and E be the points from which arcs with
a radius b will be tangent to the contour at F and M, With the
same radius and B, C, D as centers, draw arcs as shown in the
figur.e. Suppose now that P is carried around these partial areas
successively, and always in the same cyclical direction. For
(1)the point Q (Fig.58)will traverse AB, for (2),C, etc. In
each single case the record will represent the corresponding area.
Therefore the total area will be represented by the total record.
And it is readily seen that GH, IJ, KL, are each traversed twice
in opposite directions. Hence the record due to them is zero,
and the actual record is due only to the external contour. Hence,
if P were carried directly around the external contour, it should
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240 INTEGRAL CALCULUS [Note V-G.
have the same record, and hence the area. This is true for any
broken line,and hence for a curve. In the common polar planim-
eter the curve is the arc of a circle. Thus the point O in Fig.
55, Art. 88, which corresponds to the point Q moving along QX
in Fig. 58, or along ABODE in Fig. 59, moves along the circum-erence
of the circle of center T and radius a.
NOTE G
On Integral Curves
1. Applications to mechanics. " (This article is supplementary
to Art. 93.)
(a)The statical moment about OT=M^ = (area)H= ayiOH.
Hence,
M^ = ayi (aJiHX) = ay^x^ - ahy^ = a {y^x^ hyi), (1)
Also
Igh = Ipx - (area)HXf = 2 obey,- ?^^ =
ah(2y,-^\ (2)
(h)The value of HX in Art. 93 (6),ay be found by a simple
construction, though from its nature the accura"5y may not be
all that is desirable.
Let BH be drawn tangent to OB at B. Then
tanB^X = ^=-^dx HX
But 2/2-
Gidx, and hence ^=
^.hJ dx 0
Hence,
-^=^,and HX =
^^HX 0 yi
Hence, from Art. 93 (6),he point H thus determined will be
the abscissa of the center of gravity as desired.
(c)The moment of any area ORFH about any vertical XA is
proportional to the corresponding ordinate XB of the tangent
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Note G.] APPENDIX 241
to the second integral curve at the point E on the limiting ordi-ate
HF,
It has been shown in Art. 91 that-^
=
-^pFrom this,
dx b
tanZ".0: =^=^^
=2^^area,
HK dx b- ab
Hence the equation to the tangent KD is
at)
whence, aby = ab " HE-{-A(x-
OH). (3)
But from Art. 93, ab . HE is the moment of the area about
HFy and A{x " OH) is the correction necessary to transfer this
moment to an axis distant {x " OH) from HF, and therefore
distant x from the origin. The second member of (3)is thus
seen to be equal to the moment of the area about a vertical line
at any distance x from the origin. Hence, such moment is meas-ured
by aby, or ab times that ordinate of the tangent line which
is determined by the abscissa x. Hence such ordinate at any
point bears the same relation to the moment of ORFH about
the vertical line containing the ordinate, that HE does to the
moment about HF.
(d) It follows that where KD crosses OX, the moment about
the corresponding ordinate will be zero, and hence an ordinate
through K will contain the center of gravity of the area ORFH
Hence the construction given in (b)above is a special case of (c).
(e)If we apply the same proposition to the moments of the
two areas ORFH and ORPX about an ordinate through L, the
point of intersection of the two tangents at E and B, we shall
have for each moment the expressions abNL, and the moment of
the difference of the two areas or of HFPX about NS will be
zero, and therefore NS will contain the center of gravity of such
area.
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242 INTEGRAL CALCULUS [Noxa G.
Hence the tangents to the second integral curve at any two
ordinates intersect on the ordinate which contains the center of
gravity of the area of the fundamental curve lying between the
two ordinates chosen.
2. Applications in engineering and in electricity. The limits of
the present article will not allow detailed reference to the various
ways in which these curves may be made of use in studying
engineering problems. A few brief references may, however, be
made to some of the more common applications.
It is readily seen by comparison with text-books on mechanics
that if for the fundamental we take the curve of net external
force on a beam or girder, then the first integral of such funda-ental
will give the entire history of the shear from one end to
the other. Also that the second integral will give similarly the
entire history of the bending moment from one end to the other.
This serves to illustrate one important advantage of representar
tion by means of these curves, and that is, that they serve to
give not only the value at some one or more desired points, but
at all points as well.
In this way they furnish a continuous history of the variation
of the function in question, and thus give a far more vivid
picture of its characteristics than can be obtained in any other
way. In the case of beams or girde^s it may be well to note
that external forces should not be assumed as concentrated at a
point, but should rather be considered as distributed over a length
equal to that occupied by the objectto the existence of which
they are due. Thus the supporting forces at the ends of a bridge
span must not be considered as located at a point, as is common
in the analytical treatment, but rather as distributed over a length
equal to that occupied by the supporting pier. Their graphical
representation will therefore be a rectangle, or rather it may be
so taken for all practical purposes.
As another application, consider the action of a varying effort
or force acting through moving parts having inertia, and upon
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Note G.] APPENDIX 243
a dissimilarly varying resistance, their mean values being of
course the same. This is the case with the ordinary steam
engine or other prime mover operating against a variable resist-nce.
Suppose that we have plotted on a distance abscissa, the curves
of effort and of resistance. The integral of the first will give
the history of the work as done by the agent or effort,while that
of the second will give the history of the work as done upon the
resistance. Steady conditions being assumed, their mean values
will be the same. Their history, however, will be quite different.
The difference between the ordinates at any point will give the
work stored as energy in the moving parts during their accelera-ion
when the effort is greater than the resistance, restored
during their retardation when the effort is less than the resist-nce.
We might reach the same results by taking as our funda-ental
the difference between the curves of effort and resistance.
The integral of this will give the history of the ebb and flow
of energy from and into the moving parts of the mechanism.
Again, by replotting this latter curve on a time abscissa it becomes
representative of the time history of the acceleration of the
moving parts.If then the
reducedinertia
ofthese
partsis
known, the acceleration at any instant is known, and the curve
may be considered as one of acceleration. Its integral will,
therefore, give velocity, such velocity being the increase or
decrease above the mean value. Such a curve would, therefore,
show the continuous history of variation in the velocity due to
the causes mentioned.
In electrical science there are many interesting applications of
these methods. Of these only one or two of the simpler will be
here given.
Suppose that we have on a time abscissa a curve showing the
history of the electromotive force in any circuit. Then since
this is the time rate of variation of the total magnetism in the
circuit,it is evident that, reciprocally, the latter must be the
INTEGBAL CALC. " 17
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244 INTEGRAL CALCULUS [Note G.
integral of the former; Hence the first integral curve will give
the history of the total magnetic flux in the circuit.
Again^ if we have on a time abscissa a curve showing the
history of a current, then the history of the growth of the quan-ity
of electricity will be given by the first integral of such
curve.
Instances might be widely multiplied, but enough has been
given to show that where desired results may be found by one
or more integrations effected on a function whose history is
known, the complete representation of the problem naturally
leads to the production of these curves ; and for their practical
determination and for their application to many special problems,
the fundamental relations and properties as developed above and
in Chapter XII., will be found of considerable value.
3. The theory of the integraph. We will next show briefly the
fundamental theory of the integraph, an instrument for practi-ally
drawing the first integral from its fundamental. Various
forms of instrument have been devised, but in nearly all, the
kinematic conditions to be fulfilled are the same. These are as
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Note G.] APPENDIX 245
follows : Let PC (Kg. 60) be the fundamental relative to axes
OX, OF; and QD the integral relative to axes QXi, QY. For
convenience the two Faxes are taken in the same line, though
this is not necessary. P and Q are therefore corresponding
points. At Q draw a line QA tangent to QD, From P draw
PE parallel to QA. Then ;
OA dx OE a
Hence, J^ ^or y, = pcto.
dx a^
J a
If now a is constant, we shall have
1 /*" Jarea
yi = - \ydx=z or area = ayi.aJ a
These conditions are seen to correspond to (2),Art. 91. The
instrument must therefore include three points, E, P, and Q,
related as above specified. While the instrument travels along
th(B direction of X, P is made to trace the given curve, and E
remains at a constant distance a from the foot of the ordinate
through P, This determines a direction EP, and Q constrained
by the structure of the instrument to move always parallel to
EPy will trace the integral curve QD.
It is not necessary that the points E and A should lie to the
left of 0 as in Fig. 60. They may be taken as at E^, A\ and in
such case if the fundamental lies above X, the integral wUl lie
below its X as shown by QZ"',and vice versa. The actual values,
however, will remain unchanged, and the inversion is readily
allowed for in the interpretation of the results.
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Following are the figures of some of the curves referred to in
the preceding pages :
The hypocycloid, ac^+ y^ = a^. The cissoid, y^ =
2a-x
o O* X
The cycloid, x = a(^" sin "?);
y = a{\"
cos^).
246
Folium of Descartes,
aj8 + y" = 3aa;y.
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X X
The catenary, y = ^(c*+ e ").
O a
The parabola, x^ + y*= a*.
The semicubical parabola, ay^ = ic8. The cubical parabola, a!^= "*.
The parabola, r = a sec^ -.The logarithmic spiral, r = e"*.
2
247
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The curve, r = a sin' Spiral of Archimedes, r = a0.
The curve^ r = o sin 2 "?..
The cardioid, r =
a(l- cos 0).
The lemniscate, j-^ = a^ cos 2 0.
248
The witch, " = "
i^f!".
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A SHORT TABLE OP INTEGRALS
Following is a listof integrals for reference in the solution of
practical problems. The deduction of these integrals will be a
useful exercise in the review of the earlier part of the book.
GENERAL FORMULAE OF INTEGRATION
1. j(u"v"w "'")dx
= iudx " ivdx" iwdx" """.
2. Imu dx^miudx.
3 (a), iudv^uv" ivdu.
Z{b). Cri^dxuv-Cv^dx.dx J dx
ALGEBRAIC FORMS-
4. r^^^logo..J x
a;" dx =
-, whenn is different from " 1.
Expressions containing Integral Powers of a+bx
7. C(a+ bxy dx =
(^+
hxy-^"^^^^^^ ^ .^ ^iigej.gjj^ ^^ _ ^J o(n-\-l) .
249
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260 INTEGRAL CALCULUS
8. IF(Xj a + bx)dx. Try one of the substitutions, z = a-{-bXy
10.
" +
a^dx 1'" J^^ = J[K""")*-2a(a + 6a,)+a'log(a 6a;)].
dx 1,^a-|-6a?1 r dx
^^Iq
Jx{a-{-hx) a x
J x^(a'\'hx) ax or X
16. r - =^ iiog5^"^.
Jx(a+ bxy a{a+ bx) a* a
J {a+ bxf b\ a-hbx 2(a-{-bxy]
Expressions containing. a^ + a^, a^ " a^y a-i-baf, a + b7?
18. r_^^=J_iog?L""; r_^_=iiog^i^.J a^
^QcF2a a " x J a^^a^ 2a
,r-!-a
19. r ^^
=-l_tan-^a^v/-,hen a"0 and 6"0.
Ja + ftaJ* Va5^"
20. C ^^
=J-loggHLJg,
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A SHORT TABLE OF INTEGRALS 251
21. Cixf^ia-^-hotfydx
"(ripH-?n l) 6(Mi" m-f-iy
^
22. CQir{a'\-lxxfydx
np H- m-f
1 np -I-m H- !"/
23. f- *"
(m" l)aaf"-\a4-"a^/~^ (m-l)o J"""(aH-"a;")''1
(m" n+np" 1)" /* cga;
24. f-"^
1,
m " n + np" 1 /^ da;
""
an (p - l)af-^ (a+ 6af)^^ a7i (i" 1) J aj""(a+ hx^y-^'
J af
a(m" l)af-* a(m
"
1) J af"*
2g f(a+6af/(^J af
_
(a^hafy anp r(a-\-bafy-^dx
(np" m 4-l)af~^ np
"
m+l*^af
21^ r ic^diB
'"/jr6ic"/af*-"+^ a(m
" n-\- 1) T a?"*-"da;
6(m - np -f-1)(a+ bx^y~^ b(m - wp + 1)J(a+ """/
af*da;
(a4- ba^y
af-^^m + n " np + 1 (* a;*"dxIf
an(p- 1)(a+ 6a;"y-^ an (p-1) "/ (a+ 6aJ")'
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252 INTEGRAL CALCULUS
29. f" x
as*)"
2 (n - 1)a' L(a" a^-'"*" ^
" ^V
(o"+a^" J
30; f" ^J (a+ 6aO"
"^
2(n-l)aL(a6a!")"+ (^" ~ ^)J(o+
fta^"31. r_^^ = ir_^,wherez=:"'.
J(o + 6a^" 2*/ (a+ 6z)-'
32 r_^^_J (a+ bot^'
"
" "
1 C dx
2 b(n - 1)(a+ 6a!")"-"2 6(n - 1)J (a+ 6a!")"*
33. f ^=J-log
"^.
Jx(a-\-baf) an a-{-baf
Ja-^-bx' 2b^\ ^b)
3fir ^dx
_x
a r dx
37. f^^
=^log
^.
38. r ^=,i"
^r ^.
J7?{a-{-by^
ax aJ a-\-ba^
dx39
(a + "a?2)2 2a (a 4- ""a?^ 2aJ
a(a+ "a?2)22a (a4- ba^ 2aJ a + fta?
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A SHORT TABLE OF INTEGRALS 263
Expressions containing -Va + bx
[See FormulaB 21-28]
40. rxV^+todx=
-2^"ri3MVi^"M..
' loir
J^
106 6*
44. f '^'^
=JLlogV^+^-V^.fora"0.
45. r_^=, =
_2_tan-'J^"^,fora"0.
*^ arVa H- fta? V" a^
" a
46r c?a;
_ -~Va-f-6a;"_ T d
47. rV^^T^^^2V^T6^+ ar"
jJ X J
xVa
dx
4- 6a?
Expressions containing ^a? + a*
[SeeFormulae 21-28J
48. r(aj"a")ida?|V?T^" + ^loga?4-^s^T^.
49. r(a!"+o")*cte= - (2a?+ 5a")
VF+a* + ^log("+ a/^+o").
n
J w + 1 w + U
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H+2
2
254 INTEGRAL CALCULUS
H
52. ra!"(!r"a')ida!==|(2a!"a")V?Tc?-^*log(a!+V?+
"/o o
'
(to53. f '^
=log(ic+V^+T").
54. f_-^_ = /" "
-'(sr^ a*)*2 2
r_^ = llog ^=.
66. r-^^=v^+^.
56
57
5S.
69.r tto
^
Vig'+ o'
60 /dx"_
Vaj'+ g", 1
io"a+Va!'+'0'
a!"(a!"a")i2aV 2a" a;
"/Of
a;
^"a!"
62.
ri^"#^- ^^^"^+ log (a, V^+^O.
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A SHORT TABLE OF INTEGRALS 265
Expressions containing v^--^
[See Formulae 21-28]
63. f(a^-a^ida;?V?^^-f log(a; V^^^.
64. r(a!'-a")*(to=|(2!e"-6a")V5^^=^?|^log(a;+V?^."/ 8 o
n
n
66. ^x(x" a')*iB= MufQ.
67. fa!*a!"o")*to= |(2" -
a")V^^=^- 1'og (x+ V?^:^).
68. f ^". = log {x + V5^"=^").
'^ (!r"-a=)*
2
/7
da;
70. r^^=v?:r^"-
71. r-^^ = |^?3^"+ |'log(a"+V^^rp)."^ (""-ay
2 2
72. r-^^ =
-^" +log(a;+Vx^^r^')."^ (a'-a')* Var'-a''
73. f ^5_=
lsec-'*;f '^'^
=sec-"a;.
*'a!(a^-a")5" "
-^ asVa^-l
74 /^ da;_ -y/g' a'
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256 INTEGRAL CALCULUS
mmP dX
-v/aj* (3^2 \ Qg75. I irr
= jLJi"
^ 4-_:!^gec-*-.
76.J X X
"^ J(^-J-)^^^,V^+log(,V^3^')"
Expressions containing -s/a^-^oi?
[See Formulae 21-28]
78. r(a^-"^*(foj^VS^:r^+f8in-i?.
"/o 8 a
80. r(a'-a^idx ^("'-f)'-^
fCa'-aO^'dxJ^ '^
n + 1 w + lJ^ -^
11+8
81. fgCa"x^*dx
=
-("'-'^j .
J^ '
n+2
82. C"^{d'-a^idx=^(2!i^-a^-y/a^-x'J o 8 a
83. r_^_=8m-"*; f" ^"=sm-'".
rtto_
g
- (a*-a!")*'~aVa*-a^
84
85
86
-'(a'-af)*
87. r-^_=^=^_sin-
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A SHORT TABLE OF INTEGRALS 267
88.r a^-c?a. g!:^^^,3^^(m^l)aY-'^^
89. f^^
=llog ^=.
f"" ^?_=.A I t"t"_
Va^ " a*
a*a?
91. r_^^ =
_^g^+^,log
J X"
X
'-a?
93.
J(5^z;A*da,^-^5^^--'*
Qv*^ "
" ^" ^^^"
oxu." "
or X a
EXPRESSIOKS CONTAINING V2 aX " Q^, ^2 OX-\-0^
[SeeFormulae 21-28]
94, C-y/2ax-a^clx5-=it V2 ooj - ar^ + ~
vers-^^
"7.22 a
96. f^^
=yers-^g: f_^=, = vers-^ a?.
96. I af'v2aa? " a:*cte=^^
"
^"
^
m-\-2 J
97r t?a;
_
V2 ax " a^
"^a^V2aa?-aj2""
+ -
(2m - 1)oof"
i " 1 /" c?a;
i " 1)a J a?*-*2m - 1)a Jaj"-* V2aa?-aj2
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258 INTEGRAL CALCULUS
Qfi C ^^=
arW2ax"
j^(2m" l)ar oT-^dx
J"- (2m-3)aaf (2m-3)aJ a^-^
*
00. I a?v2aaj" ar^cfa;= -^---v 2 oa;" aj*-f-"vers"^-.
J 6 2 a
01 r cga; ___ V2 ax
"
g*'
"/ "V2aa;-a2oa?
02. f ^^==-V2aa;-a?^
4- gvers-^g.J ^2ax-d' a
03. r-^^^=, =
-5Jl5"V2^^Z^+
3^2^ers-^f.
*^.V2aa;-a^2 2 a
04. r^^^~^da?=V2aa;-a?" + a vers-^g"
J X a
J ar^ a; a
jjg/*V2 oa; " ar
^^ ^ _
(2aa; a^^f
07 C- ^^"
a? " a
(2aaj-af)f "V2aaj-a^
08(2aaj-ar^f "V2aa;-aj*
09. iF(x, "yj2ax"x^dx= jF(2;+a,Va^" 2*)a;,where 2=a?" a.
10. f^^
=log(a?4-a4-V2aa?4-a^.*^ V2 aa; H- ar'
11. ji^(aJ,2aa;4-ar^)^x=|F(z"a, -y/z^"a^dZyhere z=x-\-a.
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A SHORT TABLE OF INTEGRALS 259
Expressions containing a-\-hx "cai?
112.
f ^L-- ^^tan-"-|^"L,wheii6""4J a + bx + ca? V4ac-6* vToc^^
113. =
1log
2ca" + 6-Vy34^^^^^^
V6' - 4 oc 2ci"+6+V6'-4ac
6""4ac.
114. r__^_=
^log Vy + 4oc + 2ca;-"
Ja-\-bx-csi?y/h^jf.4:w: V6' + 4ac-2ca! + 6*
115. f,
^^- = ^log(2cxb + 2VcVa+W+a?).
"
"Va + bx + a? Vc
116. fVa+to + cx'da!
=^^"JVa+6a;+ca!'-^'~'*"^log(2a!+6+2 V3Vo + bx + ""*).
*"' 8 c*
117. r ^
=J-8in-".^"'-ft-."^ Va + 6aj" ex* Vc V6' + 4ac
118. CVa + bx " ca^dx
4c 8cf V6" + 4ac
119. r ^'^^
^Va
+ 6^+^_6l^g(2cirft+ 2V^Va + to + caO.
"=2c'
120.r iPtfag
_
^/a+ bx-ca^
^
b
^^^_i
2cx-b
^ Va + fix" ex*c
2 c*V6'+4ac
Otheb Alqebbaic Expressions
121- /VSJ*" V(a+x)(6+x)+(a-6)log(Va+"+ V6+5).
122. rJ|^cix = V(a -
x)(6+ x)+(o +
6)sm-"A^"|INTEGRAL CALC. " 18
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28i) VfTi"SLlL CALCULUS
"XFr"y":^TlAL AXD TRIGOXOMETRIC EXFKESSIOXS
"/loga
"/a
137. iffflx^ff, 1S9. |sinxdx ~eo6"
130. I"'/wixdx s sin;?.
131. CiAnxdx= logBecx^ " logeoBx.
133.Tcot
?/la?log
"in".
133. Cm'.x(Jxf ^^^-= log (sec? + tana?) logtan /"'+ ^\
"/J co" a?
\42/
134. Inoftpc? r/aj T-." = log (cosec " cot a?)
log tan ^"J J sin a?
^2
135. jHm!"ajrfaj"itan". 136. jcosec* a:da:= " cot a?.
137. jseotan xdx^ sec 0.
138. ronnnn a; cotirdx rs ^ 0086O 0.
130. rmu"("(fuj!!--!8in2a.J 1
2""4
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A 8H0BT TABLE OF INTEGRALS 261
140. rcos*a? cte = ^+ -. sin 2 a?.
J 2 4
141. r8in"xda.-?i5!::^^^2s^
!^^^ fsin^-^xda,J n n J
142. rco8-"(te=522r^^siM+ !L=J:fcos-'arcto.
J'
n n J
143r ^^
__
1 coso;,
n " 2 r dx
Jsin"
a? w " 1
sin*""*; n " 1 J
sin"~*a;
144 C ^^"
1 sin a; n " 2 /* da?
"/cos* a? n " 1 cos*"* a; n " 1 J cos"~*aj
145. rcos'*a?8in"a?cfa;^^^'" *a?sin*+*a?_^m---l Tcos-^^asin-acfa-.
146. I cos* a?
sin*a? da?
sin*"*in*-* a; cos^+^a;,
n " 1 /\^""^
"^,."-"^^^
1 I cos" X sm* 'X ax.
m-\-7i m + nJ
147.' ^
'"/;m"'a;cos*aj1 1
,m4-n
" 2/* dx
n " 1 sin"** x cos*"^ a?
4-n-2r
n " 1 J sii
148. f^_^Sin*" X cos* a;
___^
1.m
+ n " 2r da;
m " 1 sin**~*a? cos*"* a? m " 1 Jsin"*"^ cos" x
"
^grcos*"
xdx__
cos'"'^* a;
m^n-^2rcos"*a?da?
"/ sin" a? (n " 1)sin"~^ a? w " 1 Jsin""*"
j^gQrco8'*a;da;__ cos**-^a?
. m~l /^cos*"~'a;da?
"/ sin"a? (m "
n)sin"*a? m "
nJ sin" a?
151. fsincos" a; da? = - 525!!L".
152. I sin*a? cos xdx =sin"+^a;
n + 1
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262 INTEGRAL CALCULUS
153. rtaii-xd!r^^^^-J'tan-*"(to.
164. CeoV'xdx=:-^^^-fcoV-*xdx.... r- "
J 8in(TO+ n)a!, 8in(TO" n)ir
^^^r -, sin (m + n)a? . 8in(m
"
n)ag156. I cos mx cos nxdx==
-" ) ;" ^ +
-o^-"
-f"J 2(m + n) 2(m "
n)
^.^
/*,, cos(m4-w)ar
cos (m "
n)"157. j8mmxco8nx"to-
^^^^^^^-
^^^_^^^"
158 r ^=
,^tan-*/" tan 5\ when a " 6.
Vft"
atan|
V6+a
159. a ^ log; , when a"b
Vft*-o' VftHatanf-Vft+a
atan|6
160. f ^= tan-t "
-iwhen a"b.
J a + bsinx Vo"-6* Va*-?**
^log
, when a " 6.161.
y/b'-a"atan|+"+V6*-o"
162 r ^=JLtaii-^/^^*?5L^Y
J a*cos*aJ + fe^sin*a? aft \ " /
" A"" r -,"
J ^(a sin nx " n cos wa?)163. I ^ sm waj da? =
"A" "
^;
/'. ,
e* (sina; " cos a?)e* sm X daj= "
^ ^"
"^ii/*--" J e*^(wsinwajacosna?).164. I e^'cos nx dx = "
^^
rr^'5
/", e* (sina? + cos a?)
"* cos xdx ="^ ^
^-
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ANSWERS TO THE EXAMPLES
CHAPTER n
Art. 12
8. y" = c6"; y" = "^. 4. y" = *(a^+ "J").
CHAPTER ni
Art. 18
6* 11'
TO + n + l* 100* 22' 38*
-8 3 1 1 1 1 1
40 69fi 93fi TOX* 99x" 20a;"
6. i"t. i,", !"*,
^"^', {x^i"*
-^"-^-*\7. }x*, 2a;*, -
-?='***' **^' ***" ^**
vx
8. log^ log(" 1), I log(x"4- 3), \og(uvw+ 1),logCx"+ 2 x" - 2 x + 4),
log tan X, log sin X.
g 8"15 (m + n)"
10. " co8 2x, Bin3x, tan4x.'
log 2' log(w"+n)* 11. secix, 8in-i2x, Bin-i3x, sin-^uv.
12. tan-i2x, tan-i3x, Bec-i2x, sec-i4x, eec^a
Art. 19
11 d V2 + I
8. Jx8~x2 + 6x, Jx*-2x* + 2xi263
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264 ANSWERS
a 2 p..
- sinmx_
coaSg tan2x_cot(t"+ n)x
m 3'
2 i" + n
7.-llog("
+ ^a;"),log(4 + 3ii"),-ilog{6-23B*).o
""
J-^-'.i*--^"'-''-tArt. 20
8.K^J+V,l("+
o)^log(x+ a), 2V2ri^,
1-,A{2 + 3a;)*
5x + a
-A(3-7x)^.9. sin (x + a),
tan (x + a)," Jtan(4 " 3x), "
-cos (a + to).0
10. 2Bin|,""+*",-3e"l,-
^
2sin"x
11. A(4x-3a)(a + x)*,J_(a
+ 6x)'*(26x-3a).
a
12. log tan-i X, sin log x, " n cot- .
n
18.-L("
+ 6")",A("
+ to)*,X(a
+ 6y)".
14. i8in-i2^.15.
-Jcosec^x,isin6tf-isin*tf |Bln"^+ V"n*^+2sin^, Jtan*0
- }tan80 + tan^0 + 9tan0.
Art. 21
6.xsin-ix
+
vT319. 12. a^jLltan-i x-^
6. X COt-l X + i log (1 + "'). ," r/i NO "i . o-i"^ ^18. x[(logx)2
" 21ogx + 2].
7.a'i"
^] .
14.8intf(logsintf 1).
I log a (loga)2i 15. tanx(logtanx - 1).
'22x
.
.21 iA
"*i
8.
a"{-^--2-^^+--^}.W-
j[aoga:)"-ilogx+ H.
i log a (loga)2 (loga)8i 4
9. X tan-i X - i log (1 + x^)17. J-
c"" ("ix- 1).
10. 2C08X4- 2xsinx " x^cosx.^.w+i
/ j \
11. x^sinx + 2XC0SX - 2sinx.*
w + l\ m + 1/
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ANSWERS 265
Art. 23
6. log(2x-6 + 2Va;2-6a;). 7. log (7a; + "/7V7 x^ + 19).
8.
-^log (3a;2 + 1+ V3 VS x* + 2 x-i - 1)
2\/3
e.
ilogf^- 21.
-i-log^^^-^A
10. 58ln-if^^:^Vr 5vers-i^.^ , " ,/ ir\
V 2 y 2 22. |tan2x + logtanfx + M + a;.
"-'-"^23.
Isec-x--^-.
18. 2 log sec 8 X.
. /^^26. ilog(x2+v^^5nr^).
V6 V326. |(x2-y*)l
15. J-tan-i-^. / P
VS V5. 27. log(/3+V^ + 2V3).
16.
_l-logl^.28. J-vers-i^
4V2 y+Vs Ve ^
17.-log sin ox. 29. 4tan"i^-^"
18.-logsin((rx
+ 6). 3^ log tan ? + log sin ^.
19. JLtan-i^.1
,
VS VS 81. - log (a2x+6+o"/o'^"H2 6x+c).a
a* " 220. itan-i-^.
^^ ^^^^__^^^^^^^^^__^
Art. 24
8. ^-2x2i/~2xy2^.^4.c..a2x-^-a;j^-a%
c.
3 o "
6. ax2 + 6xy + ay2 + ^x + ey + *.
Page 55
1. "exs x-^", C^ + ^^-^-n (lyA bh^m + n + 2 \3y
2. Jx8 + 4x'--"x*
+ 16x*, iJ-H^J + ^i^.
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266 ANSWERS
8."^-2^+2a:2-8x261og(x + 2), ^+ ^r^ + 6" + 51og(" - 2),4 o 6
24 + 6 log 3.
-?^x"^, _l_a;l-^h^-^av^ 6avi.
l + " n " I 1 " " 5 4
5. " 2 cot 2 tf, f sin 2 0 + ^ cos 30, _
1 log (a + 6 cos\^).
6
6. log(y2-a2), sin a; + tan a:, iClogx)2,Pog (""+ ")?.
2a
7. 106, i, 1, 4, :, log2. 8. 2, 1, i-a,"i-"-i-
4 4 2 2 a
". i^. iog2.
-0o|2)!,?_:,l(eT_,?).
o a
10. 4\/S, ^(-l + 3v'3-2V^), 3+VI6.
12.siQ-ihtl^^ 2tan-iVTTi. 18. fx*-J"*
5?!-zi"v^f+li^.
\/l3
3 62
14. X sec~i a; " log(a; Vx^ " 1), x cosec"^ " + log(x+ Vx-^ " 1),
iccos-ia;~Vn=^, ^sin-ijck*^+ 2)Vr=^.
16. as sin " + cos X, "
.'B^cosa+ Sx^sina; + 6"cosa; " 6 8inx,
x^
X tan " + log cos x "
16. a'[^ 3^_ + _?^ ?_1 cosa;(l-logcosaj),Lloga (loga)2^(loga)8 (loga)*J^ ^ ^'
cota;(l logcotx).
17. (a;2 2 a; + 5)sin (x2 2 " + 6)+ cos (x^- 2 a; + 6).
18.(^ + "^')[2log (x8+ a8) 1]_
a8(aj5 a^)[log(x^+ a^) 1].6
19. " ^|(logx)2llogx + ?}. 22. tanx-cotx + 21ogtanx.
33.*t 3 9)
23. ^ 1 log (1 + cot 0)+ ]log (1 + cot2 6).
24. sin-^^
:"26. 0. 28. 1
sec-i^
-
1 log (x + Vx^ _
a^).
26.
rtlogtan^l
^Wftlogsintf.9. i{log(x+ VxM^)F.
27. 8in-i^^=i, sin-i^+-^.80. log sec f 4-V
V3 3 V2 4;
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ANSWEB8 267
81. log ?^2ii. 87. 14 sin-i5 - (f" + 10)VJ^^.C08" ^ 2
82. }"*vers-i-+f("+4a)V2a-". jg^ 1 logS^c^-^*
2V3 sec" + V3
83. ^V3fiI""^\og(x+y/s^"c?).'
r r"".
-^log^-4.1
log^-^ 2V5 e" + V5
1 //". "
-irx40. sec-^-^*
85. " logtan(f|V Vftloga V6
\/2 \2 o/
86.
ilogtaii(0+^y41.
2V28in-i^V2sm|42.
^tan-i
2"^+^. when 62" 4 ac,
v'4ac-6* V4 ac - 62
:log
2ax + 6-V6^^[I^^wheii6a"4ac
V"2-4ac2ax+"+V62-4ac
48.-Llog
(2oic + ft+ 2 Vo Vox^ + 6x + c).Va
44.-Lsin-^
2qg;~"^ ^ sinajcosy + c.
Va VdM^Toc
47. 3c8 + 3a;2y + 4a^2 + 2y" + c
46.-cosxcosy + c.
CHAPTER rV
Art. 29
2. (a) 74|; (6) f. 7. 251.l^- i*^'
8. (a)A;W4. ". f"/I^. 18. |.
6. 28H. 10. 24.^*- *^^^a
Art. 30
5. 81^ ir.
6. (a) ^^ir; (6) ^ir; (c)t^tjt;
iH"-
7. (a) fir;(6) 2ir; (c) ^ir; W ^ir.
8. (o) 47r; (6) iro21og(l+V2).
9. firc^xii 10. ^c*xA 11. ^- 18. t'Oft^-
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00
258 INTEGRAL CALCULUS
Oft C ^^"
^ ^V2aa? " g* (2m " l)a/* oT-^dx
J a;" (2m-3)aa;-^(2m-3)aJ x^'
a;v2aa:" ar*da;= -^-r v2aa;" 0^*4-7:vers"^ "
6 2 a
01 r__^E__ = " V2 ax"
g*
.
r g-^ "
==-V2aa?~ar'+ aver8-^g"
J"\J2(XX'-Q^
"
03. f" ^^= =
-^i^v'2^^=^
+ |a^vers-^^-'V2aa;-a^
2 2 a
04.rV2aa?-a^^^^V2aa?-ar^
+ avers-^g"Jo: a
05. rV2""z:i^,to-2^^^^E?-ver8-'2.a? X a
3aa^
Q.7r doj
_
a; -a
(2aaj-af)f "V2aic-a^
08.
r_^^_="
-^"
"^(2aa;-ar^taV2aa;-iB2
09. jV(",2ax"ix^dx= CF(z+a,Va*" 2*)2, where 2=0:- a.
10. f^^
=logfa+ a4-V2aa? + a^.
11. (V(a?,^/2ax+a^)iix=
(F(z"a, -Vz^"a^dz,here z=ix-{-a.
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A SHORT TABLE OF INTEGRALS 259
Expressions containing a-\-bx"ea?
112. rj^__
^tan-i
^^ + ^.
when6"
"4 oc.
Ja + frx+ caj* V4ac " 6* V4ac " "*
US. =
^log
2"^ + 6-Vy34^^^j^^^
Vft* - 4 ac 2ca! + 6+V6'-4ac
6* " 4 ac.
"6*
114. C ^=
1log Vy + 4ac + 2ca!-i
116. r ^
-=^log(2fla!fe+ 2-^^"+fc" + '"^-
116. CVa+bx + ^dx
*"* 8 c*
117. f^
=J-8m-'.^^-^-"^ Va + 6a;" ex' Vc V6' + 4oc
118. rVa + 6" " cai*da;
= 2^:ilV2rTto3^4.^"i^8m-'^^iiJ-.4c 8c* V6" + 4ac
119. r '^^^^
'^ Va + 6a!+ c"*
^Va+ 6"+^__6^1^g(2cr
+ 6 + 2V^Va + 6a;+ caO.2c'i
120. r'"'^
= _
^/a+ bx-cx'
^
b_
^^^_i
2cx-b
^ Va + 6a;" caj*c
2 c*V6'+4ac
Other Aloebraic Exfhessions
121. J'.^^""i(toV^+^(64^)+(a-6)log(Vo+5+V6+").
122- /"\|^"^*=(a-a;)(6+ a;)+(a 6)8m-"^^INTEOBAL CALC. " 18
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260 INTEGRAL CALCULUS
123. C^S^dx = - V(a + x)(b-x) -(a+ b)
sm-"J^^./^0 " X ^a + 0
124. yyj^-"^dx=^Vl^:r^^Qm''x.
dx
V(a?a){fi x)
125. r-^=^= = 2
8in-xfe^
EXPONENTIAL AND TRIGONOMETRIC EXPRESSIONS
126. Ca'dx=:-^^ 128. Cef^dx^^J loga J a
127. |"'daj"'. 128. |sin a?(to = " cos a?.
130. I cos a;cto = sin x.
131. Jtan a;(la; log sec a; = " log cos x,
132. I cot xdx = log sin x,
133. rsec xdx=i
" " = log (sec; + tan x) = log tan [^-f -VJ cos a?
^
\^2/
134. I cosec xdx = \-4-^
= log (cosec " cot a?) log tan -.
J J smx^^ ^ ^
2
135. Jsec* xdx=i tan a?. 136. I cosec' a?da;= " cot a?.
137. Isec X tan a^da; = sec a;.
138. Icosec a? cot a;da? = " cosec as.
139. fsin'ajda?|-
isin 2".
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A SHORT TABLE OF INTEGRALS 261
140. fcoQ^xto = ^H- ~ sin 2 a?.
J 2 4
m Mt /* "- J sin""^ X cos X
.n " l/^"""
j
141. I sm"ajdaj = "
^-=" H I sin""* a? das.
c/n n
%/
142. rcos"a;(to"""'"'^"'"'' + ?^^ fcos-'a^d*.J
'
n n J
143r "^^
__
1 cos a;,
w " 2 r cfa;
J sm"a?~ n " 1
sin**~*ajn " 1
./
sin"~*a;
144 C ^^"
^ sin a;,
n " 2 r dx
*/cos*a? n " 1 COS**"* a? n " 1 J cos**""*aj
145. fcos-a?in"a;cZa?^Q^'-^a? sin"^*x
_^m--l Tcos" *ajsin*a;da-.
J w + n m + n%/
148. Icos" a;
sin"x dx
8in"-*ajcos'"+*a; ,n " 1 rr.^""^";r.i"-i".^,*.
= 1 I cos X sm* 'a? oa?.
m-{-n m + nJ
147. f^*^"7
sm*" X cos* a?
_
_1
1 m + n " 2 r (fa?
n " 1 sin^-^ajcos""*^ n " 1 J sin'"a/'aos*"*a
148. r ^^
sm" X cos" a;
1 1,
m 4-71 " 2 r dxJ m-{-n-2 r
X cos""* a? m " 1 Jsii" 1 sin*"* a; cos""* x m " l J sin"*"'a cos" x
j^^Q/^cos"*a?da;_ cos**'''* a; m " n
+2 /*cos*"a?da?
%/ sin" a? (n " 1)sin"~* a; n " 1"/ sin""* a
150C^o^'^^dx
_
cos"*~*a? m " l /^cos*""'a?cto
"/ sin"aj (m "
n)sin""*ajm "
nj sii"
151. I sin a; cos" xdx = " " "
-5.n + 1
152. fsin"? cos xdx = 8in;:+^^J w + l
sin" a?
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ANSWERS TO THE EXAMPLES
7. }"*, 2a5* -
2jac*,^aj^,^aj 4a5i
vx
8. logf, log("-l), jlog(a"+ 3),log(tttni?+l),og(a"+ 2aja-2x + 4),
log tan X, log sin X.
9 g2x
15 (w + n)" 10. " co8 2x, sinSx, tan4x.'
log 2' log(w+n)' 11. secjx, 8m-i2x, Bin-i3x, sin-iwr.
12. tan-i2x, tan-^Sx, 8ec-i2x, sec-i4x, sec-i?.d
Art. 19
2. Jx5, ^\ ^^^, "^^, 26W.v^.11 d
\/2 + l
8. ix"-x2 + 5x, Jx"-2x* + 2x*.
26.S
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264 ANSWERS
4. 9t + 16"2 + yf" + c, ort"5(3 60" + c, a^x- {o^x* + fa***- ^x",
sin^ " cos 9.
gsinmx
_
cos3z tan2x_
cot (m + n)x
7.-i-log(a
"x"),ilog(4 + 3"8), - ilog(5-2x*).
no
..
f"in-".,itan-^,tan-^Art. 20
8. KiK+ a)*"K" +")^log(x
+ a), 2\/ST5,L.,
A(2 + 3x)*5
X + O
-A(3-7x)^.9. sin (x + o),
tan (X + a)," Jtan (4 " 3 x),
"
-cos (o + 6x).
10. 28inf,e^^ -3e-l,--
^
2'' ' '
2sm2x
11. A(4x-3a)(a + x)*,J^(a
+ 6x)*(26x - 3a).
12. log tan-i jp^ gin log x, " n cot- "
n
18.-L
(o + 6")",A
(a + jx)*.^ (a + 6v)^
14. J8in-i2"^.16.
-icosecSa;,isinS^- fsin*^ + isln'^H-ij^sin"^+2sin^, Jtan*0
- Jtan80 + tan^0 + 9tan0.
Art. 21
12.^iltan-ix-^
2 2
18. X [(logx)2- 2 logx + 2].
14. sin e (logsin ^ " 1)
16. tan X (logtan x " 1)
Uoga (loga)2^(loga)8r4
9. X tan-i X - pog (1 + x^).17.
-^
e"" (wx - 1).
10. 2 cos X + 2 X sin X " x2 cos x.
11. x^ sin X + 2 X cos x " 2 sin x.
IS. J^hos^-^\
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ANSWERS 265
Art. 23
6. log(2a;-6 + 2V"2-5a;). 7. log (7a; + V? V7 "3 + 19).
8. _"_ log (3x2 + 1 + V3V3aj* + 2a-2-l).2V3
9.
ilog^. 21.
-i-log^-^-^A-^4\/3 aj-2 + 2V3
V 2 / 2 22. itan2x + logtanfaj +
j)+ :
11.sin-i-^ "-
loo"-i"--".
y/2
23.
^sec-i0
^^^
4\/5 VE18. 2 log sec 3 05.
14. J_gin-i-X2".
V6 V326. |(x2-y2)!.
16. " tan-i *
VS VS! 27. log(/3+V/32+ 2V3).
4V2 y + V8 Ve o
17.-log sin ox. 29. i tan"! ^-i-?-
18. -log sill(ax + 6). 3Q iogtaii?+ logsin"?.
2
19.-i-tan-i^.
t,
V3 V3 81. - log (a2x+6+a Va2xH2 6x+c).a
20. jtan-i-^.32 v^^H + log (x + Vi^^^H).
Art. 24
8. ^-2xSy-.2xy2+ c. 4. a^
-^- xy^ -
x^ -{"c.
o S **
6. az^ + bxy-^-ay^ + gx + ey^-k.
Page 55
w + n + 2 \3 /
2. Jx" + 4x''-"x* + 16x^, ^J-^s.J + s.^l,
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"" ""
it, ^u '^
"^ ^^.j;u/,-
-/;^HUu fz*~"xi ^^l"v"^nF.
W. W^. ^'^
^
^'"5
6_-| co"za-logco"x),||//^// ^l'/j"/'//'U/ga/ (Wi^a/J
1#,^{C^"K";*f'Kic+ [' 22, tana; " cotx + 21ogtanx.
""' !.^
^'" n f "^/t0)-\-\\(y%(\ + cot" ^).2 2 4
Va'-*f h* a a x^
tr. Min-*'''',
Mill"'''''/^ 80. logBccf^jV
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ANSWERS 267
81. log?^5i?. 87. 14sin-i|-.(ix+10)V4^^.cos" $ 2
82. |x* vers-i-+f("+* a)V2 a-". 88.-i-log2V^ seca; + V3
83.
-Vs^"^"^\og(x+y/3^"cfi)."
89. JLlog^-A
84.^
log^-^2\/6 e" + V5
1 /v "
-ir\40. sec-1-^'
86.
-7:^ogtanf|
+
|yVSloga VS
86.
ilogtaii(0+^y41. 2V2
8in-i^V2sm|42. tan-i
^"^^ + ^ when 63 " 4 ac,
V4 ac - 6* V4 ac - ft^
"
1i^,2ax
+ 6-V6^-4(ic^when6a"4ac.
V62-4ac 2ax+ 6 +V62_4ac
43. J-log (2ax + b + 2 Vo Vox^ + 6x + c).Va
44._L siii-i-^^^:l=.
46. Binajcosy + c.
47. 3c8 + 3a;2y + 4i"y2_|.2"" + c
46.-cosxcosy
+ c.
CHAPTER rV
Art. 29
1. (a)"; W4. "" *""" ^'
2. (a) 74 J; (6) f 7. 25|.12- 1"*^-
8. (a)^^;(6)4. 8. f"/I024. w. ^.
6. 28H. 10. 24.^*- ^^''^a
Art. 30
6. 81^ ir.
6. (a) ^^^; (6)ifeir;
(c)T"Tr^;
iH"-
7. (a) fTr;(6) 2w; (c)ij^ir;(d) V".
8. (a) 47r; (6) iraMog (1+ V2).
9. ^^ch^i.10. |c*xA 11. ^. 12. i^c^.
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268 ANSWERS
Art. 32
4. y = ce"". 6. ya = to"+i + c e. cr = ^.
7.. r* = c8inn^; r=:C8in6; r = c(l"
co69).
- f (-!-)"
Page 76
1._iI"L
!""" 2a2.
^ ^ 11. 4"a".
2-
-i-12.
"a". 8,fl"8. ^/a".
3,^1
!"" ^0og8-2).*"
2 (^"ej* 3.6.7.0* 20. 2ir2(i"."
6. log4-t-^*' *"** 21. Ajiy.
.
7^16. (a) 11^; (")4a2ir2.
9. |a". 16
CHAPTER V
Art. 34
2. log*lli. 7. log(^"^)'."" a5Hlog(x+l)"(aj-4).
8. log("+ 6)2(x-7)8. 8. log[V2^:rf(x+2)].l*- |+6a5-logx"4.
-log(x-2)^(x-l).
"" i^-Sx(x^-^)''(x-p)(x + g)
10. log(a; 3)2(x-2). 16. log^^ '-^ ^.
11. log"-.
^^ x +
log|5|.6. log^^. 12. iog^-2-^A
"^
a;
a;-2 + V3 17. log V"2 +Q
"_
4.
18.TVlog"("-l)"("-2)-"(a;-8)".
^ ^
19.
.^log5.:zV^+-i_log^Ii:^.\/2 a;+V2 2\/3 x + V3
20. ilog(a;2 1)- 1 logaj(x2 2)+ Alog(a;2 4).
"""
'^'"vi%iri;7'''^"'--'^"";'-
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ANSWERS 269
Art. 35
2. log(a; l)+
-l"
9.alog(a; a)+ -^**
x+1
^
flc+ a 2(a; a)3
8. log(a;-8)?-.
10. logx(x - 1)"-?.
^2 2x4-1 2x + 2 4^x
+ 3
6. log("+ a)"(x+ 6)-" ~ 13. "
L-+ log(x+ l).
X+ 0 X + 2
6. 1 13.-_-^_
+_1_log^"^..3V6-.2-x)" 4(xa-2) Sv^ x-V2
7. log(x-3)3 ?-+ "
? 14.2^-5 2x + l
"^,iog^:zJ.'^^
^^
x-3^2(x-3)2
*
(x-2)2^2(x+l)2^^x
+ 1
^x
+ 1 x + 1 (a;-l)2 "
Art. 36
8. logx + 2 taQ-i X. 6. tan-i x + f^
,^^"
4. log(x+ l)2+ tan-ix.^^ Jlog(3x + 2)- itan-i(x+ 1).
5. ?^4.iiogx+ :5^tan-i-^.8. 31ogx+ " tan-i?^^.3*^3 V2
v^"/2
9. ^-I|!+6x log(x+ 4)"-tan-ix.3 2
10. J-tan-i-^+f" ^^" + f" ^^
11. J_tan-i-^-l.^^i+i?.
12. x + hog?^^-V3tan-i-^.6 X* V3
18. log
-^"".4-? tan-i^- Atap-i-^.
'
Vx2 + 2 2 V2 V2
CHAPTER VI
Art. 38
o"
4.^
6.-"Va?^+
q^
"Va3-x" "Vx? + a^ a^x
8X
JVq^ - xg
^(x2-qg)^
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270 AN8WEB8
8.
-i\oeit"^E"R\11.
_y^^^^^lj^/a
+v^TF\
ox
" ".-^"""fVtf^^\ 11. " i
.i^(iM^).10.
--sin-i-.IS.
^"z
Art. 40
4."* + log("*+ l).
\047o54o /
8. A"*(" + ^")*(3""-2a). 9. log(8+ 2 VSTT).
10. 35+1 + 4Vx+ 1 +41og(V"+ 1-1).
11. A(c -
")*(5x + 3 c - 24). 12. I^^"''^
.
^
(2"+3)*
Art. 41
8.
.^Vn^.4.
-^iogf^5Z?_=:^^).
V3 \VxM^+V3/
Art. 43
2.
-J-tan-if.^^^^^'^V8.
-i-log(8+ 4x + 2\^V2x2 + 3x + 4).
V2 \v^x
/ V2
4. log(x+l + V2a; + x^)X + V2 X + x"
5. Vx2 + x+l + ilog(x + i + Vx2 + x+l).-
""
-%^-"-(^2^):'"
-^---m
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ANSWERS 271
Art. 44
8. V6log(6a;-l+V6V6a;a-2x + 7).
3-i.log(6a;-l
+ 2VSV3aj2-a;+l).V3
4.
8in-i(2LlLl\6.
2V28in-i(i^^y6.
-fV6-3aj-2a;"+
-ii-8in-ifi^"^Vv^ \ \/67/
9. }V3xa_3a; + H--i?-log(6a;-3 + 2V3V3xa-3a; + l).2V3
12. iV6"a-26a; + 34 +-5^1og(5"-13
+ V5V6x2-26x + 84)5V6
+ 13 log(2x-7+V6x'^-26"34\
18.
-log(2+ " + 2Vx" + " + l\
g-114.
2ain-"(^-4^^1og(^+
^J_+^2"-^).
Art. 45
4.-V2ax-ai"+ave"-i-
"" - fx/F^^l^ + ^ sin-i 2^a 2 2 a
aaVaa-aC2a"a!" 2a" \ " /
8. - i(3a" + ax - 2 a?)V2 "" - "" + ^ Ters-i?"
". -("'' ^^)^'''--^- 10. J(ai"-2a")x^^T^.
Page 98
1. log (205+ a + 2Va;" + ax). 8. log (2a; 5 + 2 V"" - 6" + 6).
8. ^log[2V^a; - P+2NVN^^ - Pie + ^].
4. log (a -
a)(2x
- a - 6 + 2V(a; -
a)(a;6)).
5.-i_log(4x"
+ 3 + 2\^V2"6 + 3"" + 1).3V2
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ANSWERS TO THE EXAMPLES
CHAPTER n
Art. 12
8. y" = ce"; y" = "". 4. j^ = *(aj" c").
CHAPTER m
Art. 18
5* ir TO + n + l' lOO' 22' 38*
225 40 63fi 9x^ mx* 99x^ 20""
e. *x*, 4"", ""*
-^t^^\ "x^j^*,
-J?_x-?-^\+g g-P
7. }"*, 2"*.--?z, *"i ^xi ^xi 4xi
vx
8. logf, log(* 1), I log(x"+ 3), log(ttw+ 1),log(x"+ 2 xa - 2 X + 4),
log tan X, log sin X.
9 e2"
15 (m + n)' 10. " C08 2x, sin3x, tan4x.
log 2' log(w+n) 11. secjx, 8lii-i2x, sin-i3x, am-^uv.
18. tan-i2x, tan-i3x, 8ec-i2x, sec-i4x, sec-i?.
Art. 19
^11"i V2 + I
8. Jx"-x2 + 5x, Jx*-2x^ + 2x^.
26.S
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264 ANSWERS
4. 9" + 15"2 + iyt"+ c, or
,15(360* + c, a-^- faM + fa*"*-ix",
sin 9 " cos^.
A si^wa;_
cos3g tan2g_
cot (m + n)xm 3
'
2 m+n
'
7.-1-
log (a + 6a;"),ilog(4 + 3t;"), ilog(6-2"*).o
8.
^8iii-i(?,itan-i^,rtan-i^JL* 3 20 4
Art. 20
8. K" +")iK*
+")^log(a; a), 2V^T5,
i-,M^ + Zx)*,
-A(3-7a;)^.9. sin (x + a), tan (a;+ a),
" }tan (4 " 3aj),"
-cos (a + bx),0
10. 28inf,"*w., _8e-i,_.
2'" '
28m""
U. A(4a;-3a)(a + x)*,
j^(a+ te)^(26*-3a).
12. log tan"^ X, sin log x, ~ n cot- -
n
18. i(",+ 62)4,
A("+ 6x)J,
X(a+ 6y)f.
14. jgin-i"iLjZ_i.4
16.-icosec2x,
isin6^-}sin*^H-|8in8^+ V8in"^+28in^, Jtan*^- Jtan80 + tan20 + 9tan0.
Art. 21
6.xsin-ix +
Vn^.12. ?^-"ltan-i "
-^
*
2 2
6. xcot-ix + ilog(l + "^). 13 a;[(logx)2-21ogx + 2].
7 q. [g 1
] .
14. sin^(logsln^-l).Ilog a (loga)2i 15. tanx(logtanx - 1).
8.ax/_^_._i^+_^2_l
16. ^[(logx)2-ilogxi].I log a (loga)2 (loga)8i
4
9. xtan-ix-pog(l+ x*0.
H- ^e""(w""-l).10. 2 cos X + 2 X sin X " x2 cos x.
11. x* sin X + 2 X cos x - 2 sin x.
18. ^flogie L.-VOT+1 V TO + iy
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ANSWERS 265
Art. 23
6. log(2aj-5 + 2Va;2-6x). 7. log (7 a; + VT V7 x* + 19).
58.-^
log (3x2 + 1 + V3V3x* + 2a;-^ - 1).
2V3
9.
ilog^. 31.
-i-log^-^-^A-^ 4\/3 X-2 + 2V3
10.
6sin-i(-^),or5ver8-i|,;.^,2x +
logtan(x^)-f:11.
Bin-i^^.
l_.x_^a
18. 2 log sec 3 X.
14. J_8in-i:^.
4V6 VE
85. ilog(x2 + Vx*-c*).
V5 V3ae. |(x2-y2)l.
16."-tan'
" / 7^
Va V5. 27. log(/3+V/32 + 2V3).
16. JLlogJ^^^:^ 88. J-vers-i^.4V2 y + y/S
Ve ^
17.i log sin ox. 89. 4tan-i^i^a 2
18.-logshi(ax
+ 6). jq log tan ? + log sin ^.
19. itan-i2^
V3 VS 81. ilog(a2x+6+aVa^^+26x+c).(Z
80. itan-i"^. ^^ ViaZTi + log (x + Vx^ - 1).
Art. 24
8. ^-2xay-2xy2+ ^+ c. 4. aSx ^ ^- xy^ - x^y + c.
6. axa + 6xj/ + ay2 + ^a; + ej/+ *.
Page 55
8. Jx" + 4x^-|x*
+ 15x*, 7a*-H^o* + *F-
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266 ANSWERS
8.
-^-?^+2a;2-8a;251og(a;+ 2), ^+ "2 + 6" + 51og(" - 2),
24 + 6 log 3.
l+t n-lQ 6 O 4 S
l + " n " 1 1 " n 6 4
5."2 cot 2^, i8in20+ Jcos30, "
-log(a+ 6co8^).
b
6. log(y2-a2), sinx + tanar, iClogx)2,Pog("g + 6)]^
2a
7. 106, i, 1, 4, I, log 2. 8. 2, 1, l-a,".*-"-!"
4 4 2 22 a
9. je". log2,-fl2i2I",
J_i, l(ef_"?).a
10. 4VS,3Jj(_l+3v'3-2V^),
3+Vl5.
12.8in-i^+^,2tan-ivT+li.
13. Jx^-Jx*fcl" V^+T^.
Vl3 3 6-^
14. X sec"! jp _ log(x+ Vz'^ " 1),x cosec-^ x + log(x+ Vx^ " 1),
xcos-ix-Vl-x2, ^sra-ixi(x2+ 2)Vn^.3 9
15. X sin X + cos x, " x^ cos x + 3 x^ sin x + 6 x cos x " 6 sin x,
X tan X + log cos " " " "
16. a"r^3x8
^_6g
6"
1cos xd
-
log cos x),"bog a (loga)2^(loga)8 (loga)*J'^ ^ ^'
cotx(l" logcotx).
17. (x" 2x + 6)sin(x2 2x + 5)+ cos(x22x + 5).
18.^^ t ^'"^^[2log (""+ a") 1]-
a3(x'+ "")[log(x^+ a') 1].6
19. " ^{(logx)2logx + - }" 22. tanx-cotx + 21ogtanx.
o ^4^ 3 9
*
3x
23. ^ 1 log (1+ cot 9)+\og (1 + cot2 6),
L Z 4
24. sin-i^
:"26. 0. 28. 1
sec-i?
-
1 log (x + Vx-" -
o^).
26.
alogtanf|j)+"logsin^.
29. i{log(x+ VxM^)?.
27. sin-i^^, sin-i^i^.80.
logsecf^jVV3 3 V2 4/
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ANSWERS 267
81. log 5?^. 87. 14 sin-i? - (|x + 10)Vf^^.cos" ^ 2
82. |"*vers-i-+t(a;+4a)V2a-". jg^ 1 logS^^^^^^ '
2V3 secaj+V3
2 21 #"" " V6
Vs Vx + Vb1
1 q*
86. J-logtan(f|]. Vftloga Vft
V2 \2 8/
86.
ilogtan(0+|y41. 2V2sin-i ^V2sm|y
42.
^J^tan-i
^"^^+^, when 62" 4 oc,
V4ac-62 V4ac-6"
1,,^2ax
+ 6-V6^-4^when6a"4ac
V62-4ac 2ax+ 6+V6*-4ac
48.-Llog(2
ox + 6 + 2VoVax2 + 6x + c).
Art. 30
6. 81^ ".
6. (a) 4F"; W A'! WTfff'!
iH"-
7. (a) i"; (6) 2x; (c)^l^r; (d) V-
8. (a) 4ir; (6) ira"log(l+\^).
". ixc"*i*. 10. Ic**ii 11. ^. ". "-"6".
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268 ANSWERS
Art. 32
a;+ 2
8. log(" 5)"("-7)".8. log[V2^-[(a!+2)].l*-+6"-log""(*+2"
'"^-^- "--rf^- xe...iog-^-
". log?i^. IS. iog"-2-AX-2+V3 17. logVx" + ^!"!-4.
18. A loga!(a!l)"(x 2)-"(a:3)".
18._!_iog"riyi
J_log?-^
x-l
2V2 a;+V2 2v/3 x+VS
20. Jlog(a;21) Jlogfl;(a;22)+ ^ log(x^ 4).
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ANSWERS 269
Art. 35
8. log("+ l)+
-l~.
9. alog(x+ a)+
-^-r7-^-5+1 x + a 2(x + a)^
8. log(x-3) ?". 10. loga;(x-l)"--.
aj " 3 z
3 111
7.ll,_x
+ l4. log V2" + l -^^ 11. " " + ~ log^
2 2aj + l 2x + 2^4*^a;
+ 3
6. log("+ a)"(a;+6)-" ^. W. "
i-+ log(a+ 1).
X + O X + 2
" ^ 18.-_-^_ +
-i-log^+^-3V5^2-aj)" 4(x2_2) 8\/2 "-v^
7. log(x-3)" ?L.+_?
14.2x-6 2X+1
_^^x-2^
8. iog-^- +^_.
16. g_+ iog(^-^)l
x + 1 x + 1 ("-l)^ aJ
Art. 36
8. logx + 2taii-ix. 6. tan-i x + J"^" .
4. log(x+ l)"+ tan-ix..,, Jlog(3x + 2)- Jtan-i(aJ 1).
6. ^+ ilogx + ^tan-i-^. 8. 31ogx+ ^tan-i*^.
3 3 V2v^ V^
9. ^-Z^+6x + log(x+ 4)8 - tan-ix.3 2
11. J-tan-"-*" l.^*i"i".
13.
it+ilog?i"5-v/3tan-i4:.
18. log-"^"i-+;tan-i^-Atan-i-*-.
'
8. -
*.
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270 AN8WEB8
a 4 Va" + aj*\ --V2cM5-a^
8.--log
-" ^" "^^
. 11. -
ax
8.
-liog(i":^|i"Z)9
-hog/i"^iZE?Vw.
^
^ "
*"
a^V
" / "V2cM;-a^
10.--sin-i-.
IS. -"
a X oaxr
Art. 40
8. 8x* +
ilog^illiI'-V8tan-i(2xLtIVj-1 V V3 /
4. a;*+ log("*+ l).
t
m/4-4-^+t+^-4-t+"*+"*-1"i!("'947o54o /
8.fV62(a + 6a;)*(36a;-2a). 9. log(8+ 2VicTr).
10. a; + l + 4Va;+ 1 +41og(Va;+ 1 - 1).
11. A(c -
")*(^aj + 3 c - 24). 12. |^^ + ^
.
"
r2"+3)*
Alt. 41
8.-^"^Vrri5.
4. J-logf^^^:-^3
4. J_logf-^^Zzi^).2V2 \\/r=^+V2/
5. V^T6-^logf-^"IziV^\V3 Vvxnps + vS/
Art. 43
2.
_JLtan-if^^^i^^^V8. " log(3+ 4a; + 2 V2 V2aj2 + 3aj + 4).
V2 \y/2x
/ V2
4. log(x+l + V2a: + a;2)-a; + "/2" + a;^
6. Vx" + a; + l + ilog(a:+ i + Va;2 + a;+l).-
6.-2/Lz"!
2x3 ."""(va^"i).,
.,^^_.(H=j).
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ANSWERS 271
Art. 44
8. V6log(6a;-l+V5V6aja-2x + 7).
3J-log(6a;-l
+ 2V3V3aj2_x+l).V3
4.
Bin-if^y6.
2V2sin-i(i^^^y6.
-fV6-3aj-2a;2+
-ii-8in-ifi^"iV4v^ V V67 /
9. }V3xa-3x + l+-^log(6a;-3 + 2V3V3xa-3a; + l).2V3
12. iV6xa-26x + 34 +^?-log(6x-13
+ V5V6x2--26a + 34)5V5
+ 13 log /2x~7 + V6^2^-26x34\
IS.
-log(2"""2V^"iTl\14.
2sin-i(gL:^)-4V"log(^+
^J+^^^-^).
Art. 45
4.-V2ax-aj"
+ aver8-i* 5. - ?x/5a^^" + ^ sin-i 5.
a 2 2 a
-1 logf^+v^^^a8 ^V X Ie
a?Y
^^^ ~ ^^
8. - J(3o" + ax - 2aja)2(ix - x^ + - vers-i?.2 a
". -"''* ^^"^'''^^. 10. \i^-ia^)y/WT^.
Page 98
1. log (2x + a + 2 Vx2 + ax). 8. log (2x r- 5 + 2 Vx^ - 5x + 6).
8. ^log[2mx -P+2NVN^^- PbB + ^1.
iv
4. log(x -
a)(2x
- a - 6 + 2 V(x -
a)(x
- 6)).
6.-i-
log (4x8+ 3 + 2V2 V2x6 + 3x" + 1).3^
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272 AN8WEB8
e.-i-sin-iV3(x
+ l). 7."log [V5(x + 1)+ V6a;2 + 10" - 27].
8. --JLsin-i!?*?. 9. f +
JLsin-irx-n+
A "-l,
Vm " 17 17 17V2x-a^
10. ilog(2x + V4gg - 9)+
-^sin-i^^^^\
11. ~ 2 Vx2 + 2x + 4 + 6 log (a;+ 1 + Va;2+ 2a; + 4).
12. - J (2a; + llV(a; - 2)(3 -
aj)JgS^
sin-i (2x - 5)
18.
i ("-
4)Vx2-T+
}log(x+ V"2 _
1).
V2*
V aj + i y \ 2x+3 y
16. log(x + 2+Vx^ + 4x + 6)+ ^log(^^^'+^^^+^-
18. tan-iVa? - 3.
19. v/;^M^ + 21og(* + V;?+T)-Alogf3^:*"^f^+l)2 \ x+ 1 /
2^1-x 4V2 W + 3 /
21.^x*
+ a62a;a+ LV"+l^iS b^ + 2a^^) +
;^log
Vb^ + a^^ + ax).2
4a
22 xC3o2-2xg)
3a*(a2-x2)*
28.-iL "/a2
-
x-"(8x*lOa^x^ + 16a*)+^8in-i5.8 16 a
24. - Va2 - x2. 26. -
yVv'a"- x2 (3x* + 4 a^^a + So*).
26.
_llog("+^JHE^).27.
V5^3^"-alog("^t28. ^'VS^"^^ (2 x2 -
a2)+ ^sin-i^.8 8 a
29.-?-
Va2 - x2 (8x* - 2 a^x* - 3a*)+ ^ sin-i
?"
80. zpv^l^^511
_lwr""2^"Z^^x
81.
_llog(""^^
+
"?).82. ^ Vx2 " a2(2aj2" a^) ^ log (x + Vx^ " o"). 88.
^^
^"
8 8rtWxa " a^
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ANSWERS 278
84. IVxa "a^{2x^"b "")+ ^ log (x + Vx^ db o-*) S6. Vsc^To^.o 8
86. -\/2aa;-a;2 + avers-i^. 87.-J^i^^^ZZ.ox
3g,2x"
+ 6a(x + 3a)^^^__^a+6"!ver8-ig.
6 2 a
89. i^V2ax-x^ + ^%er8-i^. 40. "(3f 6ag) 3^^.,x
2 2 a 8a*("^ + "*)^ 8a6 a
41. ^ +
_l_tan-i4
48.
x + 1
x-1
4a*(x* + a*) 4a" a^
'
3(a"-x")
48. ^^^ + J__.tan-if?l:=iV4- + Hog4(x2-2x + 3)^4V2 W2 / 2 (1 -
x^)
^ ^ ^
46. Vl+x+x-^-ilog(2x+l+2Vl+x+x-^)-logf^""^+\ x+ 1 /
46. - tan-i x + 2 \/2 tan-i a/5 V3 tan-i
-v/|.2 '3
47 nin-12 a'^*' ("'+ "')(a' + fe" "") 4. :f"i
(a"-6")(a"+ 6"-a!")'
4
61.2x + 6
^^^2 ^_ 4a. ^_ 3 ^ |iog(2g + 1 +\/4x-^ + 4x + 3).
62. 5-=^Vx2 + 2xH-3. 63.
--^V- 8 + 12x-9x2-||sin-i(3x-2).
2 lo
54. "(2to + m)Vte.4.^4.^ + ^^-4Zn^.^,W^jx + m_\
4^ 8iV-Z V\/m2-4in/
551
w r(" + V^^^="^)' + (3 - 2 V2) q2 .
2aV2L(x
+ Vx'*-o'^)*^ (3 + 2V2)a2J
gg2V3xg-2x + l
2x-l
CHAPTER VII
Art. 46
6. (o) sin X " 8in" x + ^ sin*" " | sin"'x ;
(6) " cos X + J cos'x " i cos* X ;
(c) " cos X + cos8 X " I cos* X + ^ cos^ X.
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274 ANSWERS
(6)-
Jsin6xco8x+
jfsin*x"to,see
(a)];
(c)-?i5!"."2i^-rsin"xda;,
see(6)].7 8
"/
8
(6) Jsinajcos^x+ ircos^xdc,see(a)].
(c) j8inajcos7x+|fcos6x(to,see(6)].
8. (a)-li2|^
+ ?f-rV-"seeEx. 4).^ ^
4 8in* 05 4 J sm" jk
(6) i secx tanx + logVsec a; + tanac
(d)li5M.+ 3C_^, [see(6)]."
-' 4cos"ie 4J co8""
"" ^ ^'
U)liiH^
+ lf-^. [8ee(c)].'-'5 cos's; 5 J C08*ac
"" "^ '""
(/")_1.52?"_"cotx
+ c.^"'^
3sin"x^
(J,)_i^
+ |f^, [8ee(/)]."^
Ssin^x 5^ sm*x
Art. 48
4. (a) Jtan8x4-tanx + c. (c) - icot^x- f cot'x - cotx.
(6) - Jcotsx-cotx. ("i)tan8|3tan|+c.6. (a) Jtanxsecx + ^logtanf^iy
(6) Jtanxsec8x + I|tanxsecxlogtanfj+ |H.
(c)-20t^^!"""+ilogtanf. x\
(^) " jcotxcosec^x " { (cotxcosecx " logtan- !"
(6) ftanfxsecfx +
flogtan^j+iyArt. 49
2. (o) itan"x-tanx + a;. ("0 - i cot^ x - log sin x.
(6) Jtan*x-itan2x + logsecx. (e) - Icot^x+ cotx + x.
(c)itan6x-Jtan8x + tanx-x. (/) - J cot* x + i cot2 x + log sin x.
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ANSWERS 276
Art. 50
2. I sin^a; "
T"j-sin3 a; 4- ^y sin
'"x. 3. " ^ cos* a; + A cos
*x.
4. - 2 Vcos a; (1 " I cos^ x + i cos* "). 6. |sin'x(l " ^sin^x + i sin*x).
6. J tanfi X + itan* x + e. 7. " } cot^ ^ " i cot^ a; + c.
Art. 51
2 cosa;/8in^a; sin^x sinx\ .x - cosg
xqp^pgyN
^^
2^3 12 8 J'^ie"
2sinx^^
2'
POS X ^
3. tanx " 2 cotx " 4 cot*x. 6. ^" cosx " | log tan*.^
2sin2x
^ ^
2
Art. 52
2. itan2x + 21ogtanx-icot2x.^ _
cosec^x
^ ^ ^^^^^.^ __ ^ ^^^^^^^
y
3.T^^tan-Vx
+ ftan^x. $. Jtan^x + ^tan^x.
4. " J cot" a; " I cot* x. 7. 4 sec* x (^gsec^ x "
^^rsec x + J).
Art. 53
2. i/^-sin2x+^^y 8. |x + isin2x +uVsin4x.
4.Y^^ (5x " 4 sin 2x + J sin'2 x +
-}in 4
x).
6.3iy(5x
+ 4sin2x " Jsin82x+ |sin4x).
6.
_sin^J^^^^_sm4x.7.
^i^/sx-
sin4x +
"""^'
(3tan?-2)
4.
itan-i(2anfV1.
-Ltan-i? ^ ^^
/- K 1 1""tan X + 3
3
_i_log2tanx
+ 3-v^.5. ilog^-"
^.2V5 2tanx + 3 +
v^
3.
ltan-ifitan?V6. Jlog^^^IL^V 2/
^ ^
tanx + 2
Art. 56
1. i6'(sinx-cosx).4- Ac-3"(2sin2x-3co8 2x).
o 1 ,/"
.. \
" 5. " ie-*(sinx + cosx).2. J
e'(sm
X + cos
x).
2 v i^ y
" V A ,
^,/i,2sin 2x + cos2x\
3-TV"^'(3sin3x
+ 2cos3x).""i"'^l ^
)"
INTEGRAL CALC. 19
48 16 64 """V 8
Art. 55
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276
I _
cos 8 a; cos 2 x
16 4
" sin 1 1 X. sin 3 X
^*"l2~'^~6~*
CHAPTER VIII
Art. 59
4. 240. 5. 80.
6. Tlie double infinity of straight lines y = cix + Cj, in which ci, cj are
arbitrary constants.
7. 3y = 2x(x2-l). 8. i(n^ -
mfi)(d c)(6-
a). 9.ir(/3 o).
Art. 69
8.Tab. 4. 5ir2a".
Art. 70
8. 2ir^a^b. 6. jira2^. 7. (t - ^)a'^h.
4. jm. 6. 4} cubic feet. 8.irVpqh^.
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ANSWERS 277
Art. 71
4a2a
= VSSTT^^ + a log ^^L+^+a.Va
8 = 2.29558 a.
5. s = 4
a(cos
^" cos
^J;
length of a complete arch = 8 a.
6. 8 = 6a.
Art. 72
3. 8a. 4. "r"?,vT+^-^iVnMr2 + log??-"^4:"^l-
vr2(a+Va2 + ri2)J"
7. 2a/V6-2-V31og^ + "^ 1.
I v^(2+V3))
=
atan^8ecalogtan(^j^;" =
2a(sec+ logtanf
rV. s
Art. 73
1. s = r0. 2. s = 4a(l cos^"
3. (1) " = 4a(l "
COS0); (2)8 = 4asin0.
4. (1) "=ptan0sec0 +plogtan (^+ jj;
(2) 8=p tan (0+1]sec^0+|Wi)logtan|+^]pV2-i)log2p\/2.5. 9* = 4a(sec80 " 1). 8. s = clog sec 0. "
6. , =
"logtan(|l). ^ . = ^8in.0.2
7. " = a(e"* 1).
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278 ANSWERS
Art. 74
8. 2xb /"+ "
-cos-^^Vor 3. }iraJ(a+ x)*'
V Va2-
62 ay, ,.
2ira6fVl-6"+?^5::i^V^. "/
V e r 5. V'"'-
in which e is the eccentricity.^'
*"(*" 2)a^.
7. ^^Tra^
10. ^-1",or 32.704".
11. in".
5. fa.
6. I a.
Page 164
1. fro*.
a. Volume = J7ra25;surface = 2ira2 + " log?-"-^,in which e is tlic
eccentricity.^ "" ^
8.
x{(x+|)v5rnr2--fiog^^+
^+^^^^^^"+^}.
4. 4ir2a". 6. 7r6\/a2 + 6i. 6. 4ir2a6.
7.
faft^cota.8.
ia%.9. Volume = Y- a^ ; surface = 8 a'^. 10. | ira*.
.
11.y*5 ira^.
12. Volume =
2"a"(^^'""g^"^'''-ec08.);surface = 4
ira^(sin" a cos
a).
13. Volume = ir^a' ; surface = V ^^^2. 14. Surface = ^ ira^.
16. Volume =
^^8^" -
?V surface = 8
)ra2(rj).
le: 1^(10-3ir).
17.3J5ir(3a2
4a6 + 8 62)A.6
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ANSWERS 279
18. 2ay/S. 19. 8=Vb^-\- y' + b log^^'' + ig-ZlJ,here 6=
--^.y log a
22. logVS. 28. ^. 94. ?^. 96. ^[W + 4)i-8].
CHAPTER X
Art. 79
5. The density at a pohit three fourths of the distance from the vertex to
the base, namely, ^kh.
6. x = ih, 7. 2c = }^; x = }A.
8. x = y = }a. 9. x = 6. 11.
x=|^y = 0,
19. Mass = f *a", if density = jfcistance.
Mean density =
.4244max. density.
Center of mass is at y = 0, S =
^^ ra =.680 a.
18. ^ = 0,x =25"in". 14. a8 =
i",F=16.
o a OT OT
15. (a)x = ih,y = 0;
(b)x = ih, y = ik, in which ji;s the ordinate corresponding to as = ^.
17. x = y = JJf.^. 18. x = y = ia. 19. x = 4a,y = 0.
90. X ="
I a, 1/= 0. 94. At a point distant f a from the base.
91. x = }",y = 0. 96. ic =
-|a,y= 0.
Art. 80
(Inthe answers M denotes mass)
4. If a is the radius, I=iMa*j k =
-^.
y/2
5. /=:"M"i"J^. 6. A;=
-^.7. ^.
12 V6 20
8. (o)/=i3f62; (6)/ = J3fa2; (c) / = Jif (a2+ 6^).
9./=3f^.
10. /=i3f(62 + c"). 11. /=J3fa2. 19. I=iMa\3
18. /= i^a" = i 3fa2, since 3f = f jfca*by Ex. 12, Art. 79.
15
14. I=Mm., u =^.
/=m,W^ + f\ or 3f(^%n.
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280 ANSWERS
CHAPTER XI
Art. 82
2 62.4.9 2.4.6. 13
4 ?+l a;^.1 ' 3 a"
,1 " 3 . 6 x^
*
7 2* 6 2.4 11 2.4.6 16*
6. }x*(l-ia;2-^VaJ*-ifVa^ +^-)
+ c.
^' * \,m^ q m-Hn^ 1 . 2 . g2 w + 2"+ j^^'
an^/:j:rzfi
.
1 sin^x .
1.3 sin* x,
\,^
". logx + x + ^^ +
3-^^+...+c.
3|3^6[5 7[7^^
Art. 83
^23^2 4 5^ 2-4...2n 2 n + 1
6.
log(l+x)=f-||-^+....
log(l-x) =
-^-^-^-^.^ ^12 3 4
^V^
2 3 2-4 5 2.4.6 7
CHAPTER XIII
Art. 97
Art. 98
2. yVl -x-^ + xVl -y^= c. 8. tany = c(l
-
e*)".
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AN 8 WEBS 281
Art. 99
. 2. a;jfa c2(a; 2y). Z. y = c^.
Art. 100
3
8. xV + 4jB8y-4xy8 + y"-xe"' + c2"y + a5* = c.
Art. 101
a. 2alog" + alogy-y = c- 8. "a-i^-l = "a5.
4. xV + my2 = cx*.
Art. 102
2. y = (x+ c)"-". 4. y(x2+ 1)2= tan-la; + c.
1
8. y = x2(l+ C"^). 5. ""y = 0x4-0.
Art 103
a. 7y"* = ex*- 3x". 8. y^ =
c(l-
x^)*-i^-
4. 60 y"(x+ 1)2= 10 x^ + 24 x^ + 15^* + c.
Art. 104
2. 343(y+ c)8= 27axL 8. (y -
c)(f/ x2.- c)(xy+ C2^+ 1)=0.
Art. 105
a. log(i)-x)=
-^+c, with the given relation. 8. x=logp2+6p+c.p" X
Art. 106
1.y=c-alog(i)-l),x=c+alog^.
.
y-c=V^^-U,n-^^-
Art. 107
8. j/= cx + sin-ic. 4. j/=
cx+^.6. y^^cx^ + l+c.
Art. 108
8. The catenary j/= ^(e"e~")r
^= cosh-i-.
4. The envelope of the family of lines y = ex +
-^,namely the parabola
(x-2/)2-2a(x + t/)+a2= 0.
^
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ANSWERS 283
Art. 114
-4t
4. y = cie-2* + e'(c2os V3 x + Cs sin \/3 a).
6. y = cie-' + C2e -2* + csc^*. 6. y = Ci6-*" + C2C"* + cje*.
7. y = ci6*" + C2e-*" + Cs sin(ax + o).
Art. 115
2. y = ci + e-*(C2 cjx). 3. y =
e-*(ci+ C2X + csx^) cie**.
Art. 116
8. y = Cix"i + C2X-2. 4. y = x\ci + C2 logx).
6. y = x2[ci+ C2logx + C8(logx)"].
Page 227
2. y2 = 2xy " + x". 12. y-"+i=ce(" i)"n*+2sinx+-^.dx n"1
8. tanx tan y = k^, 13. JL= ^^ + 1 + cc*'.
iV.y
4. x" + y^ = 2anan-i|c. ,^^ ^^^^ ^^^^^. ^^^^,^^,
6. xy{x"
y)=c.
6. ax2 + 6xy + cy2 + grx + ey = ife.l^- 2y=cx2 + ^.
c
7. x = cye^.16. y2_.2cx + c2.
8. x2 + " = c.
y 18. (cix+ C3)2+ a = c,y".
X 1""
^^Y^^a^^^' ^^' y = "Ji + "^2" + cse** + C46-
10. y = tan X - 1 + c"-t"".^0. y = Ca - sin-i cie-".
11. y = ax-\-cx Vl - x^. 21. y = Cie-* + C2 + |e'.
X
222. y = Ci"-* + "2(
2 cos " X + Cs sin -
23. y = c sin (nx +o)(ax H- 6).
24. y =
"'(ci+ C2X)sin x + e*(C8+ C4X)cosx.
25. y = ci + C2X + e*(c8+ C4X).
56. y =
x(cicoslog X + C2 sin log
x) + CsX"!.
27. y = (ci+ C2 log x) sin log x + (cs+ C4 log x) cos logx.
28. J?/y =
|^?|!-|J4-CiX
C2..
29. j"/y =
-7"-v-+cix3-fC2X2 + C8X + C4.
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INDEX
[The numbers refer topages.]
Algebraic tran'sformatious, 103, 100, 107,
.
108, 113.
Amsler, 188.
Amsler's planimeter, 189.
Angles, use of multiple, 114.
Anti-derivative (anti-dififerential),, 5,
7, 11, 12, 14, 21, 25.
Applications to mechanics, 167.
Approximate integration, 177-189.
rules for, 181-188, 2:i6.
Areas, change of variable, 141, 142.
derivation of integration formulae
for, 9, 27.
oblique axes, 140.
polar coordinates, double integra-ion,
139.
polar coordinates, single integra-ion,
135.
precautions in finding, 63.
rectangular coordinates, 58.
rectangular coordinates, double in-egration,
126.
rules for approximate determination
of, 181-188, 235-237.
surfaces of revolution, 152-156.
surfaces z"/(a;,y),
156-160.
AnziHsrjr aquaUon, 223, 224.
Bernouilli, James and John, 2.
Bertrand, 237.
Boussinesqt 186.
Cajori,2.
Gardioid, area, 138.
center of mass, 173.
intrinsic equation, 151.
length, 149.
orthogonal trajectories,16.
surface of revolution, 156.
Carpenter, 189.
Catenary, area, 76.intrinsic equation, 160.
length, 147.
surface of revolution, 156.
volume of revolution, 77.
Center of mass, 168.
Change of variable, 41.
Circle, area, 61, 139, 140, 141.
evolute of, 166.
intrinsic equation, 151.
length, 146, 149.
orthogonal trajectories,17.
Cissoid, center of mass, 173.
length, 149.
volume of revolution, 77.
Clairaut, 213.
equation of, 213.
Complementary function, 222.
Cone, center of mass, 172.
moment of inertia, 175.
surface, 164.
volume, 71, 73, 143.
volume of frustum, 77.
Conoid, volume, 143.
Constant of integration, 22.
geometrical meaning of, 23.
two kinds, 25.
Convergent, 178.
Cotes, 236.
Curves, areas of, oblique axes, 140.
areas of, polar coordinates, 135-140.
areas of, rectangular coordinates, 58.
derived, 29.
derivation of equations of, 26, 26, 75,
134, 214-217.
integral, 3.3, 190-200, 240-245.
intrinsic equations of, 149.
quadrature of, 58.
Cycloid, area, 141.
intrinsic equation, 151.
285
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286 INDEX
Cycloid, length, 147.
note on length, 144.
volume of revolution, 142.
volumes and surfaces of revolution,
166.
Cylinder, 143.
moment of inertia, 176.
Density, 167.
Derived curves, 29.
Derivation of equations of carves, 26,
26, 76, 134, 214-217.
Derivation of fundamental formulse, 48.
Differential equations, see Equation.
Differential, integration of total, 62.
Differentiation under integration sign,
190.
Durand, 186, 187, 190, 237.
Equations, homogeneous in x, y, 206.
linear, constant coefficients,223.
linear, homogeneous, 226.
linear of firstorder, 208.
linear of nth order, 222.
linear, properties of, 222.
of order higher than first,218-228.
reducible to linear form, 209.
resolvable into component equations,
210.
solutions, general, 202.
solutions, particular, 202.
solvable for x, 212.
solvable for y, 211.
variables easily separable, 206.
Evolute, of circle, length, 166.
of parabola, length, 165.
Expansion of functions in series, 179.
Exponential functions, 117.
Figures of curves, 246-248.
Fisher, Irving, 30.
Folium of Descartes, area, 138.
Formulae, areas, 9, 27, 137, i:"9, 141.
lengths, 145, 148.
of approximate integration, 183, 185,
186, 187, 235-237.
of integration,fundamental, 37,47, 48.
of integration, table of, 249-262.
of integration, universal, 39, 44.
of reduction, 46, 93, 94, 95, 101, 102,
106,110,231-234.
surfaces, 154, 155, 157, 168.
volumes, 70, 130, 143.
Fourier, 9.
Fractions, rational, 78-83,.229-231.
Functions, irrational, 84-99.
trigonometric and exponential, 100-118.
Geometrical applications, 68-77, 126-166,
214^217.
meaning of constant of integration,
23.
principle, 7.
representation of an integral, 14.
Graphical representation of a definite
integral, 73.
Gray, 189.
Gregory, 180.
Hele Shaw, 189.
Henrici's report on planimeter, 189.
Hermann, 188.
Hyperbola, area, 68.
orthogonal trajectories,17.
related volumes, 77.
Hyperbolic spiral, area, 138.
length, 149.
Hypocycloid, center of mass, 172, 173.
intrinsic equation, 152.
length, 147.
surface, 156.
volume of revolution, 76.
Integrable form, 37.
Integral, complete, 203.
Integral curves, 33, 190-200.
applications, 195-193.
applications to mechanics, 196, 240-
242.
applications to engineering and elec-ricity,
242-244.
determination of, 198.
relations, analytical, 192.
relations, geometrical, 194.
relations, mechanical, 195.
Integral, definite, 8.
definite, evaluation by me.asuring
areas, 181.
definite, geometrical representation,
14.
definite, graphical representation, 73.
definite, limits, 9.
definite,
precautions
in finding, 67.
definite, properties, 15.
definite, relation to indefinite, 24.
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INDEX 287
Integral, elliptic, 147, 180.
general, 23.
indefinite, 22.
indefinite, directions for finding, 54.
multiple, 120.
name, 1, 21.
particular, 23, 204, 222.
Integrals, derivation of, 48.
fundamental, 36-38, 47.
table of, 249-262.
Integraph, 198, 200.
theory of, 244.
Integrating factors, 207.
Integration, aided by changing variable,
41, 141.
approximate, 177-189, 235-237.
by parts, 44, 46, 100, 231, 233.
constants, 22, 202.
definition, 1, 18, 21.
derivation of reduction formulae, 231,
233:
fundamental formulae, 37, 47.
fundamental rules and methods, 36.
in series, 177-179.
mechanical, 188, 189, 200.
of a total differential, 52.
precautions, 63, 67.
sign, 2, 21.
signs in successive integration, 125.
successive, one variable, 119-123.
successive, two variables, 123-125.
universal formulae, 39, 40, 44.uses of, 1.
Intrinsic equation of a curve, 149.
Irrational functions, 84-99.
Jevons, 30.
Lamb, 237.
Laurent, 180.
Legendre, 180.
Leibniz, 2, 59, 144, 180.
Lemniscate, area, 137.
Lengths of curves, polar coordinates, 147.
rectangular coordinates, 144.
Limits of a definite integral, 9.
Logarithmic curve, area, 76.
length, 165.
Logarithmic spiral, area, 137.
length, 149.
Markoff, 186.
Mass, 167.
Mass, center of, 168, 169.
Mean value, 17, 160-164.
definition, 163.
Mechanical integration, 188-189, 200.
Multiple angles used, 114.
integral, 120.
Neil, 144.
Newton, 2, 69, 73, 144, 180, 236.
Oliver, 4.
Orthogonal trajectories,14-217.
Parabola, area, 5, 14, 59, 68, 76, 126, 138.
center of mass, 173.
derivation of equation, 26.
intrinsic equation, 151j152.
length, 146, 149.
length of e volute, 165.
orthogonal trajectories,16, 217.
semicubical, area, 68.
semicubical, intrinsic equation, 152.
semicubical, length, 144, 147.
surfaces of revolution, 155, 164.
volumv of revolution, 71, 77.
Parabolic rule, 184, 186, 236.
Paraboloid, center of mass, 173.
volume, 131, 143.
Pascal, 59.
Planimeter described, 188, 189.
theory of, 237.
Pyramid, volume, 143.
Quadrature, 58, 73.
Range, 161, 163.
Rational fractions, 78-83.
decomposition of, 229-231.
Reciprocal substitution, 84.
Reduction formulae, 46, 93-95, 101, 102,
106, 110, 231-233.
Roberval, 58.
Signs of integration, 2, 21, 125.
Simpson, Thomas, 184.
one-third rule, 182, 184, 186.
three-eighths rule, 236.
Slope of a curve, 25.
Solids of revolution, surfaces, 152-156.
volumes, 69.
Solutions, general, 202.Sphere, surface, 155, 158.
volume, 131, 132, 133, 143, 164.
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BY THE SAME AUTHOR
DIFFERENTIAL EQUATIONS
AN INTRODUCTORY COURSE IN DIFFERENTIAL EQUATIONS FOR
STUDENTS IN CLASSICAL AND ENGINEERING COLLEGES
334 pages. $ 1.90
'The aim of this work is to give a brief exposition of some of the devices
employed in solving differential equations. The book presupposes only a knowl-dge
of the fundamental formulae of integration, and may be described as a
chapter supplementary to the elementary works on the integral calculus/" Extract from PrefacfS.
In use as a text-book in Johns Hopkins University, Baltimore, Md. ; Van-
derbilt University, Nashville, Tenn. ; University of Missouri, Columbia, Mo.;
Purdue University, La Fayette, Ind. ; Wesleyan University, Middletown, Conn. ;
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