formula list math 109

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



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



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



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)

formula list math 109

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