formula list math 109
DESCRIPTION
Integral CalcTRANSCRIPT
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NECES Academics Committee Stephanie Grace de Guzman
Math 109 Formula List|A.Y. 2014-2015
FORMULA LIST
PROPERTIES:
1. du = u + c 2. (du + dv dw) = du + dv dw 3. Rdu = du c
POWER FORMULAS:
xndx = xn+1
n + 1 + c if n 1
x1dx = lnx + c if n = 1
undu if n -1:
undu = un + 1
n + 1 + c
if n = -1:
undu = lnu + c
EXPONENTIAL FUNCTION
audu = 1
lna au + c
eudu = eu + c
TRIGONOMETRIC FUNCTIONS
1. sin u du = cos u + c 2. cos u du = sin u + c 3. tan u du = ln sec u + c
= ln cos u + c
4. cot u du = ln sin u + c = ln csc u + c
5. sec u du = ln(sec u + tan u ) + c 6. csc u du = ln(csc u cot u) + c 7. sec2u du = tan u + c 8. csc2u du = cot u + c 9. sec u tan u du = sec u + c 10. csc u cot u du = csc u + c
TRIGONOMETRIC TRANSFORMATIONS
I. sinmx cosnx dx where m or n is a positive odd integer tools: change the one w/ odd powers sin2x = 1 cos2x cos2x = 1 sin2x
II. secmx tannx dx or cscmx cotnx dx a. Where m is positive even integer
tools: sec2x = 1 + tan2x csc2x = 1 + cot2x
III. tannx dx or cotnx dx where n is an integer tools: tan2x = sec2x 1 cot2x = csc2x 1
IV. sinmx cosnx dx where m & n are positive even integers
tools: sinx cosx = 12 sin2x
sin2x = 12 (1 cos2x)
cos2x = 12 (1 + cos2x)
V. sin ax sin bx dx
sinmx cosnx dx
sinmx cosnx dx
tools: sin sin = 12 [cos( ) cos( + )]
cos cos = 12 [cos( ) + cos( + )]
sin cos = 12 [sin( ) + sin( + )]
INVERSE TRIGONOMETRIC FUNCTIONS
1. dua2 u2
= Sin-1 ua + c
2. dua2 + u2 = 1a Tan-1ua + c
3. duu u2 a2
= 1a Sec
-1 ua + c
ADDITIONAL FORMULAS:
1. u2 a2 du = 12 { u u
2a2 a2 ln |u + u2a2 |} + c
2. 22
= ln|u + u2a2 |} + c
3. a2 u2 du = 12 { u a
2 u2 + a2 Sin-1 (
)} + c
4. duu2 - a2 = 12a ln | u - au + a | + c
5. dua2 - u2 = 12a ln | u + au - a | + c
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NECES Academics Committee Stephanie Grace de Guzman
Math 109 Formula List|A.Y. 2014-2015
HYPERBOLIC FUNCTIONS
1. sinh u du = cosh u + c 2. cosh u du = sinh u + c 3. tanh u du = ln |cosh u | +c 4. coth u du = ln |sinh u | +c 5. sech2 u du = tanh u + c 6. csch2u du = coth u + c 7. sech u tanh u du = sech u + c 8. csch u coth u du = csch u + c
IMPROPER INTEGRALS I. Integrals with infinite limits in the integrand
*in other words, isa or both a and b sa formula
na b
af(x)dx, infinity.
af(x)dx = limb
b
af(x)dx
b
-f(x)dx = lima-
b
af(x)dx
-f(x)dx = lima- and b
b
af(x)dx
NOTE:
&
00 = pag ganyan yung situation, dun sa
equation/s kung san naka substitute yung b or a, derive both the numerator and the denominator. Then you may start dividing
1 = 0
II. Integrals with infinite discontinuities in the integrand *in other words, isa or both a and b sa formula
na b
af(x)dx, pag sinubstitute sa f(x)dx,
UNDEFINED yung lalabas. a) If f(x) increases numerically without limit as x a, then
n
mf(x)dx = limam+
n
af(x)dx
a) If f(x) increases numerically without limit as x b, then
n
mf(x)dx = limbn-
b
mf(x)dx
a) If f(x) increases numerically without limit as x c, a < c < b , (kumbaga yung point of discontinuity,
hindi given pero nasa gitna siya ng a and b) then,
b
af(x)dx =
c
af(x)dx +
b
cf(x)dx
= limnc- n
af(x)dx + limmc+
b
mf(x)dx
INTEGRATION TECHNIQUES/PROCEDURES/METHODS
I. Integration by Parts
udv = uv vdu
WALLIS FORMULA
*only works when the upper and lower limits are 2 and 0.
2
0sinmxcosnxdx =
[(m-1)(m-3)2 or 1][(n-1)(n-3)2 or 1](m+n)(m+n-2)(m+n-4)2 or 1
where: = 2 , if both m and n are EVEN
= 1, if other wise
II. Substitution Methods
A. Substitution of Functions
example: x 1 + x
u = 1 + x
x = u 1
dx = du *then substitute sa mga x
B. Algebraic Substitution
example: x 1 + x
u = 1 + x
u2 = 1 + x
x = u2 1
dx = 2udu *then substitute sa mga x
C. Reciprocal Substitution
use them for: 2++
Substitute: x = 1y dx =
dyy2
D. Trigonometric Substitution
If you see this combination: Substitute these:
a2 u2 u =asin
a2 + u2 u = atan
u2 a2 u = asec
2ax - x2 x = 2asin2
2ax + x2 x = 2atan2
x2 - 2ax x = 2asec2
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NECES Academics Committee Stephanie Grace de Guzman
Math 109 Formula List|A.Y. 2014-2015
E. Half Angle Substitution
z = tan12 (nx)
dx = 1n
2dz1 + z2
tan(nx) = 2z
1 - z2
sin(nx) = 2z
1 + z2
cos(nx) = 1 - z2
1 + z2
III. Partial Fractions
A. Linear & Distinct Factors
dxx(x - 1) = dx B. Linear & Repeated Factors
dxx2(x - 1)2 = dx C. Quadratic & Distinct Factors
dxx2 + x + 1 = A(2x + 1) + Bx2 + x + 1 dx yung imumultiply sa A, aka yung 2x + 1, is yung derivative ng dnominator
D. Quadratic & Distinct Factors
dx(x2 + x + 1)2 = dx AREAS AND CENTROIDS OF PLANE AREAS
A. Vertical Element
A = (ya yb)dx
Ax = x(ya yb)dx
Ay = (ya2 yb2)dx
B. Horizontal Element
A = (xR xL)dy
Ay = y(xR xL)dy
Ax = (xR2 xL2)dy
ANALYSIS OF POLAR CURVES
I. Symmetry
ox: F(r,) = {F(r , -)
F(-r, - )
oxy: F(r,) = {F(r , - )F(-r , - )
ox: F(r,) = {F(-r , )
F(r, + )
II. Intersection w/ the pole
set r = 0 and solve for i
III. Intersection with axes
IV. Critical Points
set drd = 0 and solve for C
V. Divisions
use i & C
VI. Additional Points
SOME COMMON POLAR POLES
A. Limacons : r = a bsin or r = a bcos
0 < | ab | < 1 with a loop
0 < | ab | = 1 cardioid
1 < | ab | < 2 with a dent
| ab | 2 convex
0 90 180 270 360
r
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NECES Academics Committee Stephanie Grace de Guzman
Math 109 Formula List|A.Y. 2014-2015
B. Rose Curves
r = asin(n) r = acos(n)
VOLUMES AND CENTROIDS OF SOLIDS OF REVOLUTIONS
A. Method of Circular Disk
V = b
ar2dh
Vx = XCdv
Vy = YCdv
CONDITIONS:
1. element must be parallel to the axis
2. r must be parallel to the axis
3. the axis should be a boundary
B. Method of Circular Ring
V = b
a(R2 r2)dh
C. Method of Cylindrical Shell
V = 2 b
axydx
(when using a vertical element)
V = 2 b
axydy
(when using a horizontal element)