Formulas for integration

∫du/u

ln(u)+C

∫f(x)-g(x)dx

∫f(x) - ∫g(x)

completing the square

(Ax²+Bx+(B/2)²) - ((B/2)²+C)

for integrals involving an expression in the form √a²-u², where a is a positive constant and u is a function of x….

u=asin(θ), then express everything in terms of θ

integral of √a² +u²

u=a•tan(θ)

integral of √u²-a²

u=a•sec(θ)

∫ du/u²+a²

= 1/a tan^-1(u/a)

integral of two functions being multiplied

use formula: uv-∫vdu

where we can easily obtain du

special integration of  √ a²+ u²

factor 

• outside integral (1/#)

• inside multiply function by #

find u, a, du, 

then solve with formula 

½ [u•OG + a² ln|u+og)] + C

when we have integral of sin³x•cos⁴x 

focus on the ODD power

• factor using trig identities

get it to a form with ∫ u du

• use power rule

when we have ∫ cos⁴ x dx

express in terms of cos or sin

• (cos(x))⁴ 

• (cos²(x))² 

apply reduction formula

expand 

take out constants

continue doing that 

trig identities (sine and cosine) 

sin²x+cos²x=1

trig identities (tangent and secant) 

tan²x+1=sec²x

∫sec(u)du

ln|sec(u)+tan(u)+C

∫sec²(u)

u’tan(u)

∫csc(u)

-ln(|csc(u)+cot(u)| + C

but also 

ln|csc(u)-cot(u)| + C

reduction formulas cos²x

(1+cos(2x))/2

reduction formulas sin²x

sin²x= (1-cos(2x))/2

cos(−𝜃)

cos(θ)

sin(-x)

-sin(x)

cos(A)•cos(B)

½ [cos(A-B)+cos(A+B)]

sin(A)•sin(B)

½ [cos(A-B) - cos(A+B)]

sin(A)•cos(B)

½ [sin (A-B) + sin(A+B)]

cos(A)•sin(B)

½ [sin(A+B) - sin(A-B)

partial functions (distinct linear factors)

factor denominator

set as separate additions with constant as numerator

numerator= A times whatever doesn’t cancel put from 2factors/factor for A)

repeat for b

expand

solve coefficient by coefficient

repeated linear factors

add a partial fraction for each (increasing in degree)

ex.

x(x+1)(x+1)

A/x + B/(x+1) + C/(x+1)²

quadratic irreducible 

write linear at the top in form

Bx+C/quadratic