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BC Calculus | Post AP: Advanced Integration Techniques

I. Integration by Parts / Reduction Formulas If )(xfu = and )(xgv = and if /f and /g are continuous, then ∫ ∫−= .vduuvudv Example 1: Evaluate ∫ .2 dxxe x Example 2: Evaluate ∫ xdxx 2sec Example 3: Evaluate ∫ xdxln Example 4: Evaluate ∫ − dxx1tan Example 5: Evaluate ∫ .22 dxex x Example 6: Evaluate ∫ .cos xdxe x Example 7: Evaluate ∫ .sec3 xdx Example 8: Find a reduction formula for dxxn∫ sin Example 9: Use the reduction formula in Example 8 to evaluate dxx∫ 5sin


Integration for powers of trigonometric function (Reduction Formulas):


II. Trigonometric Integrals ∫ dxxx nm cossin Guidelines:

Example 1: Evaluate dxx∫ 5sin without using “by parts.” Example 2: Evaluate ∫ dxx2cos without using “by parts.” Example 3: Evaluate ∫ dxxx 43 sincos


III. Trigonometric Integrals ∫ dxxx nm sectan Guidelines:

Example 1: Evaluate dxxx∫ 53 sectan . Example 2: Evaluate ∫ xdxx 42 sectan .


IV. Trigonometric Substitutions Expression in Integrand Trigonometric Substitution

22 xa − θsinax =

22 xa + θtanax =

22 ax − θsecax =

Example 1: Evaluate ∫− 22 16 xx

dx

Example 2: Evaluate ∫+ 24 xdx

Example 3: Evaluate ∫−x

dxx 92


V. Integration using Partial Fractions Guidelines for Partial Fraction Decompositions of )(/)( xgxf

1. If the degree of f(x) is not lower than the degree of g(x), use long division to obtain the proper form.

2. Express g(x) as a product of linear factors bax + or irreducible quadratic factors cbxax ++2 , and collect repeated factors so that g(x) is a product of different factors of

the form nbax )( + or ncbxax )( 2 ++ for a nonnegative integer n.

3. Apply the following rules.

Rule a: For each factor nbax )( + with 1≥n , the partial fraction decomposition contains the sum of n partial fractions of the form

,)()( 2

21n

n

baxA

baxA

baxA

++⋅⋅⋅+

++

+ where each numerator kA is a real number.

Rule b: For each factor ncbxax )( 2 ++ with 1≥n and with cbxax ++2 irreducible, the partial fraction decomposition contains a sum of n partial fractions of the form

nnn

cbxaxBxA

cbxaxBxA

cbxaxBxA

)()( 2222

211

+++

+⋅⋅⋅+++

++

+++ , where each kA and kB is a real number.

Example 1: Evaluate ∫ −−− dxxx

x32

352 .

Example 2: Evaluate dxx

x∫ +

+2)2(

76

Example 3: Evaluate dxxx

xxx∫ −−

−−−32

3422

23


VI. Integration using Table of Integration

Example 1: Evaluate ∫+

dxxx 22 53

1 for 0>x

Example 2: Evaluate ∫ dxxx cos3


VII. Irreducible Quadratic Expression in the Integrand

Example 1: Evaluate ∫ +−− dxxx

x136

122

Example 2: Evaluate ∫++

dxxx 258

12


VIII. Miscellaneous Substitutions (Integral contains an expression of the ( )n xf )

Example 1: Evaluate ∫+

dxxx

3 2

3

4

Example 2: Evaluate ∫ +dx

xx 3

1


IX. Miscellaneous Substitution (Integrand is a rational expression in xsin and xcos )

Example 1: Evaluate ∫ −dx

xx cos3sin41

Example 2: Evaluate ∫ +dx

xx2sin1

cos

482 llll CHAPTER 7 TECHNTQUES OF TNTEGRATTON

- ^1 l

lll. l--dx*Jzx'-7

fax13. l ,--. dxJ x--bx

1^,14x'-lx'-4

15. l-dxJ3 x' - 2x'

x-1_ )-ûLx'*3x12

1_ )*(x+a)(x+b)

x3-4x-70

-dx

x'-x-6x2+2x-1_)-

-Y' - -T

x'- 5x + lb

e2'_ )-e"*3e'*2

COS -I_ )-..1 qi

sm:x t sln x

sec2/

tan2t+3tal t+2e'

(e'-2)G2'+t)

'r. lJ

,0. I

l6'Jn

,t. J

,0. j

,,. T

,o.l

,,. 7

,t. J

or.J

or. J

ro. JLlx l,

'r. J

,,. T

,r. j

Ej,,.

T

dt

4v2-l.v-12" )-.y(y+2)(y-3)

1_ )-(x+5)'z(x-1)

x3+4

-dx

-I L y',

ds--r-------l-s-l,s - 1l'

x2-x*6

-dx

x'*3x

x'+x+1,

-AX

(x' -t 1)'

7^x--ZX-l_ )-(x-l)'G'+1),3x2+x+4l-dxJ x -i3x-t./.

ft xl-dxJo x'44x*13

I -J- a,J-r'*1

, x4+3x2+ll-dxJ x'* 5x'* 5x

rx3+2x2*3x-2I _ )*J (x'*2x*2)'

dx

dxdx (2x+7)(x-2)2 5 l-52 Use integration by parts, together with the techniques of thissection, to evaluate the integral.

sr. J

m(x'? - x + 2) dx i?. I.tan-txdx

ffi sl. Us" a graph of f(x):ll|' - 2x - 3) ro decide whether

lif G) a* is positive or negative. Use the graph to give a rough

estimate of the value of the integral and then use partial fractions

to find the exact value.

ffi Sl. Craph both y : 1lG' - 2x2) and an antiderivative on the- same screen.

55-56 Evaluate the integral by completing the square and using

Formula 6.-r

x*4l79.ll"--dx-J x'*2x15

-r 1

l3l.l l-dx-Jx'-1

rt x3+2x33. l---dxJo.r*+4x'*3

rdx35. I .=-) x\x'l- 4)'

)^r x'--1x+ I17, I =---dxJ (x'-4x*6)'

-t dx

155.1 I ----_J x'-2.x r"j

4x2 + 12x -'72x-l 1

36.

57. The German mathematician Karl Weierstrass (1815-1897)noticed that the sribstitution t : tan(x/2) will convert any

rational function of sin x and cos -r into an ordinary.rational

function of r.

(a) Ifr: tan(xlz), -r I x ( .r, sketch a right triangle oruse

trigonometric identities to show that

39-50 Make a substitution to "xp{ess

the integrand as a rationalfunction and thbn evaluate the integral.

/r\ 1cos[;):--- and\L/ VI L

(b) Show that1

-,21Lcosx:-:------, and\+t'(c) Show that

2)-

-

-

)+uL- ûL1+t'

58-61 Use the substitution in Exercise 57 to transform the inte'

grand into a rational function of f and then evaluate the integral.

rdx58. l-J 3-5sinx

60. l't' I d]J.B 1 + sinx-cos,

/r\ t,tr\r/:ffi

.2tstn.r:

-1+t'@jqo.

116 Ê41. I dxJg x-4

-r x3

WJ I -;-dxr Jx.+ I

rdx40. l-J2Jx*3ix

rr 147. | . , .1-dxJU I _i_ VX

"" r:qq.l' !^ axJtlz x' * x

[rrinr; Substitute u : $i.]r1I ---=------;-r dxr \/x-lx

r ^/l * ./xl-d*rx

45.

3sinx-4cosx

Multivariable Calculus O9l0: Unitl - Principles of Integrations (Unit Exercises)

Principles of lntegral Evaluation

A convergent improper integral over an infinite interval canbe approximared by first replaciag the infinite limit(s) of in_tegration by finite iimit(s), then using a numerical integrationtechnique, such as Simpson's rule, to approximate the integralwith finite limi(s). This technique is illusffated in Exercises60 and 61.

60. Suppose that the integral in Exercise 5g is approximated byfirst writing it as

f+* " rK r*a| ,-" dx = | "-r' d* + I e-,' dxJo Jo Jx

then dropping the second term, and then applying Simpson,srule to the integrai

tKL-" d,

Jo

The resulting approximation has two sources of error: theerror from Simpson's rule and the error

r*aE: I e-,'dx

JK

that resuits from discarding the second term. We cal E thetruncatian etor.(a) Approximate the integral in Exercise 5g by applying

Simpson's rule with 2n: lg subdivisions to the inte_raIb^q

t3

L-" d*Jo

Round your answer to four decimal places and compare

_ . it to j.,8 roundeO to four decimal ilu."..(b) Use the result that you obtained in Exercise 46 and thefactthate-xz S lxe-* forx > 3 to show that the fun_cation error for the approximation in part (a) satisfids0<E<2.1x10-5.

61. (a) It can be shown rhat

[+- 1I

-d*:!Jo r0+1 3

Approximate this integral by applying Simpsou,s rulewith 2n : 20 subdivisions to the integral

Round your answer to three decimal places and com-pare it to n/3 rounded to three decimal places.

G) Use the result that you obtained in Exercise 46 aad thefact that ll@6 + | < l/x6 for x > 4 to show that thetruncation error for the approximation in part (a) sads_fies0<E<Zxl0-4.

62. For what values ofp does [*- "rdx converoe?lo

63. Show ,au, ['4 "oru".g"rifp < 1 anddivergesifp > I.Jo xp

64. It is sometimes possible to convert an improper integral iatoa "proper" integral having the same value by making anappropriate substitution. Evaluate the following integrat Uymaking the indicated substitution, and investigate whathap_pens if you evaluate the integral directiy using a CAS.

f4 1I

-dxJo xb+7

ffi

al

J,u=$-1

65.

66.

/,1 cos xJ, "n

dx; u: Jx

ft sinxI --_dx;u-Jl-xJo t/ I - x

I *-x:- dx;

1. Consider the following methods for evaluating integrals:a-substitution, integration by parts, partial fractions, reduc_fion formulas, and trigonomefic substirutions. In each part,state the approach that you would ry first to evaluate theintegral. If none of them seems appropriate, then say so.You need not evaluate the integral.

{e) | an-'*a*t_(l

.l ,Jq - x2 dx

r_(D

JJt-xzdx

2. Consider the following trigonometric substitutions:

x:3sin6, x-JtznQ, x:3sec0In each part, state the substitution that you would fy first toevaluate the integral. If none seems appropriate, then statea trigonometric substitution that you would use. you neednot evaiuate the integrat.

f_@ I ,/g*x2dx @) l6-dxr "J"

O, | ,sinxdxf_(c) ltan'xdx| ^)I -7X'le) I

-dxJ x'*7

(b) /"ou sinxdx

fal It^'xsec2xdx

rrt 1ff6a.

at ! ,E-u'0,

at I Jr*u'0.[ ,t.'- ro.

f-

J Jt + (ex)z dx

(d)

(f)

14. Find apositive value ofa Lhat satisfies the equation

rt' iI

-dx:lJo x/*az

3. (a) What condition must a rational function satisfy for the

method of partial fractions to be applicable directly?

@) If the condition ia part (a) is not satisfied, what must

you do if you waat to use partial fractions?

4. What is an improper integral?

5. In each part" find the number of the formula ia the Endpaper

Integral Table that you would apply to evaluate the integral.

You need not evaluate *re integral.

@ | swxcos 9x dr Al I A' - xsles' dx

15-30, evaluate the iategral.

t7.

19.

I /cos 0 s:Jr-? d0J

f^I xtanz(xz) sec'7x'} dx

fdx,_J {3 + *ryrt,f cos0t --- )aJ sia'0-6sa0*12f x-13I

-

dv

I Jxz +2x +2fdxI @-1)(x*2)(x-3)

rc. !0"/o

t^r' e ae

rrlJl18. / 1t -2x213/2 dx

J -tt Ji

(c)2t.l .J' - .'o'

| "o'a*6. Evaluate tne int.fa

/1(a) integration by parts

fdx,_I xJ4x *3

I'#0.

(b) the substitution u: JVfi.7. In each part" evaluate the integral by making an appropriate

substitution and applying a reduction formula'

fal I snauax *, I ,coss1*2;dx

zs. ['F o, zG.J+ x

27. [ -La. zB.J Je'*lr+- x dx,,.J, @T*1'+- dx

'o' /. ;,-TW, a.b > o

I#,'Id,I xiz*x*1)['Go*J6 x*9

^Ia2I J*-ta*Jo

i--"-'-I Some htegrals that can be evaluated by hand cannot be eval-

i

! uated by all computer algebra systems. fu Exercises 31-34, i

I evaluate the iategral by han4 and determine if it can be eval- i

I uated on your CAS. !

L .-.- -... .. - ---- ..-- - I

(d)

(f)(e)

#*;t-1i,:

ii

,fi

i.:ii.l

,l'.ii.'.,i

'i):

r1:

I

I r.'

l'.r

flConsider the integrat I -;- dx.- J xt_x(a) Evaluate the integral using t}le substitution x : sec 6 '

For what values of x is your resuit valid?(b) Evaluate the integral using the substitution x : siaO'

For what values of .r is your rcsult valid?(c) Evaluate the iategral using the method of partial frac-

tions. For what values of x is your result valid?

(a) Evaluate the iategral

[-Lo,I Jx_xzthree ways: using the substitution u : G, using the

substitu[on a - JT1, and completing the square.

(b) Show that the answers in part (a) are equivalent.

Find the area of the region that is enclosed by the curves

) = (x - 3)l (x3 + xz), y -- 0, x : 1, andx : 2-

11. sketch the region whose area r, l*- , .-4" and use your

sketch to show that

[* d' : [' aio,lo l+x'z - Jt V

U. Find the area that is enclosed between the x-axis and the

curve y : onx - l) / xz for x Z e.

13. Find the volume of the solid that is generated wheu the re-gion between lhe x-axis aad the curve y : e-' forx > 0 isrevolved about the y-axis.

(a)

(b)

(c)

m31. 1#,.Eil 32. / t"or" x sioro r - cos30, stn32 x) dx

,:----:B 33. J,l. - J*z - +ax.lfiint: +(Jx +2- 4E -Dz :?l

m 34' I #*dx.fHint:Rewritethe denomi:rator as

x10(1 + x-e).1

ffi 35. Let

f(x): -zx5 +26x4 * 15x3 + 6xz +aox + 43

x6 -xs - 18xa -2r3 -39x2 -x -20Use a CAS to factor the denominator, and then writedowu the form of the partial fraction decomposition.

You need not filrd the values ofthe constants.

Check your answer in part (a) by using the CAS to findthe partial fraction decomposition of /.Integrate / by hand and fhen check your answer byintegratirg with the CAS.

36. TheGarnmafunctian, f (x), is defined as

1+of (x) : 1 (-te-l dt

Jo

It can be shown that this improper iategral converges if andonlyifx > 0.(a) Find f (1).(b) Prove: f (x * 1) : xl(x) for aII .x > 0. [;1inr: Use

iategration by parts.l(c) Use the results ir parts (a) and (b) to find f (2), f (3),

and f (4); and then make a conjecture about I-(n) forpositive integer values ofn.

(d) Show that f (+) : ,/i.fHint: See Exercise 58 of Sec-tion 8.8.1

(e) Use the results obtained in parts (b) and (d) to show thatf (i) : i"fi *dr(i): ifi.

Refer to the Gamma function defined i-u Exercise 36 to showthat

rl(a) / (t *)' dx = (-l)"1(n * 1), n > 0.

Jo

fHint:l*tr: -Inx.](b) [** ,-,' a, : p(' + 1),

n > o." Jo \ n )'Vlint: Let t -- xn . Use the result i:r Exercise 36(b).1

Asimple pendulznz consists of a mass that swings in a ver-tical plane at the end of a massless rod of length .L, as shownin the accomFanying figure. Suppose that a simple pendu-lum is displaced through an angle 06 and released from rest.It can be shown that in the absence of friction, the time Irequired for the pendulum to make one complete back-and-

where 6 : 0(r) is the angle the pendulum makes with thevertical at time l. The improper iategral in (1) is difficultto evaluate numerically. By a substitution ouflined below itcan be shown that the period can be expressed as

frvn 1

':oli J, mrado Q)

where k : sn(00/2). The integral in (2) is called a com-plete elliptic integral of the fvst kind arrd is more easilyevaluated by nrrmerical methods.(a) Obtain (2) from (1) by substituting

cosd : I -zsnz@/z)cosdq: I -Zs1n2@o/2)k : sin(00/2)

and then making the change of variable

sin d : sin(0 /2)lsn(0012) : slr:(O /2)/k

O) Use (2) and the numerical integration capabiliry of yourCAS to estimate the period of a simple penduium forwhich I : 1.5 fr 0o : 20o,and g - 32ftl s2.

5t.

ffi s8'

forth swiag, calTedthe period, is given by

- [t,l.rt" I'r- l-l :)aY s Jo /cosE - cos A; -- (1)

Railroad Designour company has a contract to construct a track bedfor a railroad lirue between towns A and B shown on the contour

map in Figure L The bed can be created by cutting trenches through the surface or by usirLg some combination oftrenches and tunnels. As chief engitleer, your assignrnent is to arlalyze the costs of trenches an"d. tunnels and to proposea design strategy for minirnizing the total construction cost.

sw Engineering RequirementsThe Transportation Board submits the following engineering requirements to your company:

. The track bed is to be straight and 10 m wide. The grade is to increase at a cotrstant ratefrom the existing elevation of 100 m at town A to an elevation of 110 m at point M and thendecrease at a constant rate to the existing elevation of 88 m at town B.

A7,ffi

,.l"dv:uv-lrn,

,./.o, udu:sina*c

'. I *, + b). dx:'Tii'ii' . r, n * -l

,. | *r," + b)^ dx - @x +-!)*11+* - #). ,,

r. [ *t"* + b)-t dx : L - 4rrbx + bl + c

ll,n ,,, a v11--! I9')xhx+dzdx= o,L or*)*c

f

-

21{or46y+z11. I (\'ax -f b)" dx:----* * C, n* -2-- J' a n-t2

The formulas below are stated in terms of constants a, b, c, m, n, and so on. These con_

stants can usually assume any real value and need not be integers. Occasional limitationson their values are stated with the formulas. Formula 5 requires n * -1, for example, andFormula 11 requires n * *2. The formulas also assume that the constants do not take onvalues that require dividing by zero or taking even roots of negative numbers. For exam-pie. Formula 8 assumes a * O, and Formula 13(a) cannot be used unless b is negative.

z. I o. a,:

a. I sin, du

e.[(ax+u)

!*c, a*1, a>olna

:-cosz*C

1-tdx-_Llnlax+bl-Ca

n* -7, -2

n. I d* ,:1rn l--z l * .J x(ax+b) b lax+bl

n. luG-tb d*:2rGr+a + ul--!-J x J x\/ax+b

f dx r l\,6-+b-{Ot151 |

-:---lnt

-t+C.

if b>0' )xYax-tb \/b lYax-rb+Ybl

,0. l{r--)odx: {,r+n ol dx

J x' -iJrt6o-c

,r. l___!_=_{4*u _a l---4!-- ^Jx2\/ax+b bx -2b)*{o*+b -L

rc. I ^d* ^:lt*-, L+ c) a'lx' a a

,r. l--lt-:l,n1x+ql+cJa'-x' ./.a lx-al

zo. l-L: sinh-l L + c: In (x + lE + *') + cJ\/o2+*2 a

xlx r {^n r_+c'Zaz(az a *21 ' 2a3 'o a

xlfa* I-

2a2(a2 - x2) 2az J az - x2

t7. l--!1-- -J \a' t x')-

tg. [ ' ^d* ,,^ :) la' - x')'

Section A7 A Brief Table of lntegrals 613

f---2zr. ) f a, + xl dx:;\/Cr+7 + ir"1* + tG\ xzt + c

t- ^4zz. ) rt7 + x, a* : !la2 + z*'){7 + *, - f r, 1* + t/il +?7 - c

f \/E+? t o+{oz+*2 I23.J _ dx:\/aztxz-atnj , l+c

26. l--L:-1 ,nlo*@l*,J x\,/az+xz a I x I

x. [-L: sin-r L+ cJ \/a2-x2 a

I t7:7 lx \F-pJx'axr ' r ta*f;'1_ x'zl

34. l-!!-:-rml"___]_:_____^l+cJ x\,/a2-x2 a I x I

f-z3. | __.:_ ax:

J \./a2 - *2

3s. l=-L) x2\/ az - xz

* I #: cosh-r X* r: h l, +!* - o,l+ c

f-_^2tt.

J t/-, - r, a. : |t/-, - r, - Tr"l* + f*, - orl + c

38. I(\/F-,,)'a*:N#L #l(f xz- az)'-'zax. n* -l

L (f *z - "21'-z40. ) x\x'z - o')' dx: \jL + c. n * -2

f- ^4u. J *17 - r' ax: {ox2 - oz1{*z - oz -}m l* + {*' - *l + c

r v?-, r.rlz.

) . dx:lx2-a2-asec I ;l*,

r,)1 I d* --ffi -.

J x2\,/ a2 + xz a'x

29. I t1r, -7 d*:.x t f^z .: , oz ,--J ,va'-*-* 2"n'L+C

a2 tx I ^r.Isint:-1xYa2-x2+C

^/-v a'- x'1 +C

a'a

_ sin-, l=:).,*. | --:4-: a sin-r 1.u-) - yTax - x2 + cJ 1/2o*-*z \ a I

la*lJ x\/Zax - x2 a

ur. / "o, ax dx: L sit ax + c

sl. /.or' ax dx : t *'# - ,

oo. Irin, axdx:- sin'-laxcosax +n- ] f .rr,- 2axdxJ na n J-

ur. [.or, ax dx: cosn-l ax sin ax * n - 1 |) nu n J

cos'-z ax dx

I cos(a + b)x cos(.a - b)x

f sllr.,a - b\x sin (a * b)x{b)J sin axsinbxdx:-r; _ U --1G +U + c, a2+ b2

I sin (a - b)x stn (a -t b)x(c)JcosaxcosbxO*: ,**b) + ,1o-y, +C, az+b2

It56. I sin ax dx : -1.or ax * CJa

f x sin 2axss.Jsin2axdx:i--;-,


63.

65.

66.

68.

69.

70.

7t.

n1

74.

75.

76.

78.

80.

82.

84.

86.

88.

90.

/ ,in n, cos d.r a* : - 99!2!* * 6 f sin'-lax64. I sinn axcosaxiv: -, - 5. n * -lj (n* L)a

l+u- d*: I rn,sin axl -r cJ SI1AX A

I "osn-t

axI cos'ax sinaxdx: - i*"',--" t C, n * -lJ (n-t t)a67. I tin o* dx: -! ln l.o, axt + C

J COSdx a

I sinn -t ax cos'-t ax n - | L .^ "I sinn ax cosm ax dx : -

Jffi-;i)sinn_2axcoS*Qxdx,n*-m(Ifn_-m'useNo.86.)

f ,in,o, cos*axo*_ sin'-tqxc,os^-tax -+f ,in,rrcos,-2 axdx, m*-n (ffm:_,n,useNo.g7..1

J atm+n) m-rnJ

Irr*^: "+-,,,-, [8,,"(+ -+)]-r, b2,c2

I a* -t lr+bsino*+\7r2-b1 coso*l ^ ,, - a

)n-rt't"*:;ffiInl b*csinax l-t' o-<-c'

Ii*:-:,*(; -T)., ,' Ir:k;:L,un(+. +)- ,

I or : --Z:,un-, I E* gl * .. bz > czJ b+ ccosdir a\/b2^c2 LYa+c 2J f

I a* I lc+bcosax+\F-b2sinaxl) t -;a,*: hr'1ff |

+ c' b1 < c2

I o- : l,un !'J l*cosax o Z+C

,r. I o* : -l ,or9! + cJ l-cosax a '2

I * rino* a* : \sin ax - L cos ax + c)a'a

D. I * cos ax dx: 1.o, ax + Lsin ax - c)a-a

f ,' ,t,, ax dx : - t "o, ,. * :l xn-. cos ax ctx il.

| *" cos ax dx : t sin ". - :[ xn-t sin ax dx

[ ,un o* a* :! tn lsec axl + c ss. J

.ot ay dx :1tn lrin axl + c

Irun'o*a*:!tanax*xtCss.

J "ot' ax dx: -! "orax

* x t C

| ,un' o* a* : 'T'-t o,! - f ,un' 2 ax dx. n * IJ a(n-I) J

sz. [.ot, axdx:_ co:r'-ta.! - f .or,,-, axdx, n*1J aln_t) J

/ ,"" ,, a* :! ln lsec ax t tan axl + clt

89. I csc axdx : -a1n lcscax - cotaxl * CJa

/ ,".' ax dx: L tan ax + c lr. /"r"' axdx: _.L"otax* C

92. I sec'ax dx

%. I csc' ax dx

94. I sec'axtan C,

_ secn-z ax tan ax La(n - l)

cscn-z ax cot ax: -- a(,r - I) -

- sec'axaxdx:-+Cna

n-21I sec"-z ax dtN*IJ

n-21* - I csc''-z ax ,n-lJ

n*0

, n*l

dx, n*1

es. /",.

tz. l,o,

lo3. I bo'dx: lT * r, b> 0, b + lJ a lnb

ro5. I x,eo* dx: 1 *, n IJ a "--;)xn-leaxdx

- cscn ax''axcolaxdx:---+C, n+0na

I-l axdx:ircos-t ax - !\/l - a2x2 - Ca

r98.

J tan-1 axdx: xtan-lr. - *h (1 + azxz) + C

ss. | *^sin lax d*: J::sin-rax -; r[ #, nr -lJ

,*. /x, cos-r axdx: #cos-,cx . r=l i#, rt*,-1

ror. /x,ran-,

axdx:#*"-, o,-fi1 # n t -l

rc2. leo*dx:leo,-CJa(f .n,

704. I xe'- 4x::, (ax - l) - C

106. i x,bo, dx: *nbo* , IJ alnb-

'rr=b)xn-tbaxdx) b>o' b+l

fir. fe" sin bx dx : # (a sinbx- b cos bx) + C

,*. / eo* cos bx dx : ;# (a cos bx* D sin bx) + C

I xn-t(ln ax)^ m f110. J

x'(ln ax)* dx: --#i- -;; I xnfin ax)*-t dx,

f lln ax\'-l111.

J x-r(lnax)*dx:-; n, + C. m* -l

109. I lnax dx : xlnax - x * CJ

n* -1

f,

w. | -:!-: tn ltn axl + cJ X tlrAX

fi4. Icosh ax a*: lsinh ax * CJaf sinh 2ax x

116. J

cosh2 ax dx : ff -, + C

n*0

113.

115.

r17.

ltJ

sl*axdx: -coshax*C

t . " sinh2z

J sinh'zaxdx:,2;a-i*,

/ ,mn, ax dx : ry - + I sithn-z ax dx,


+C

inh ax dx

n*0

'S

C,

coshnaxo*- coshn-t axsirthax +n- | lcoshn-2axdx, n*0nanJ

xsinhax d*:lcoshax -{sint ax* C 120. Ixcoshaxd*:Lsinhax-{"ort

x'sinh axdx:4.orh o*-Ll xn-t coshaxdx 122. lxncosh axdx:4rirt or-Ll *'a aJ ) a aJ

ftanh ax a* : Ltn (cosh ax) + c 124. I coth ax a, : lln lsinh axl + caJa

tanhz ax dx : x- 1,* ax * C 126. I cothz ax dx: x- 1 coth ax * Ca-Ja

tanhn ax dx : - t?nh'-! .ax + | tanh,-z ax dx, n * 1

ln-l)a J

coth'- ' c

cothn axdx: - -..--',.o* + | cothn-z axdx, n * I

ln-l)a )

sech ax a* : Lsin I (tanh ax) * C 130. [ "r.r, ax dx :f m l,unn 9' I

o ,frax)*C J a I Tl*,

sechz ax d* : I tanh ax -l C .jJ2. I cschz ax a* : -L coth ax * Ca-Ja

sechtr ax o* - sech'.-z ax-tanh ax + " - 2, I s"rh'-2 o* d*, n * I(n-l)a n-l)

csch'-2axcothax n-21cschraxdx: - _ --------= | csch,'-'axdx, n* I

\n-l)a n-LJ

sech'axtanhax dx:-sechnax *r, n*o 136. Icschnaxcothaxdr:-"t"ho* +nqJna

eo, sinh bxdx: +l-L - '-0. 1* ,. az + b2z 1al_b a-b)'-'

.o*I I

e*cosh bxdx:'l_^ |'o' . +'-",1 + C, az + b22la+b a-b)* *,-rr-, dx: (n - l)1, n) 0 ,*. f c-n,' dx : +rE a) O

t 1.3.5 "' (n- l)'ff ifnisaneven

118.

tt9.

tzL.

123.

L2s.

127.

128.

L29.

131.

1.33.

t34.

135.

t37.

138.

t39.

I

I

I

I

I

I

I

I

t

I

I

t

I

I

Iirteger >- 2

if n is an oddinteger > 3

2.4.6'..n 2'cosn x dx : I Z. 4.6 ... ln _ 1)lo''' "n.

* o* : ln'''

3.5.7 ..-nt{t.

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