Integration techniques

integration by parts

∫u dv\= uv - ∫v du\###integration by trig sub...

√a² - x²

substitution: identity:

x\=a sinθ 1 - sin²θ\=cos²θ\###expresison:

√a² + x²

substitution: identity:

x\=a tanθ 1 + tan²θ\=sec²θ\###expression:

√x² - a²

substitution: identity:

x\=a secθ sec²θ - 1\=tan²θ\###integration by partial fractions

1. factor the denominator

2. seperate the expression based on factored form

3. instead of a constant in the numerator, put a variable (capitalized)

4. solve for the variables

5. once solved for variables, substitute them back into original factored expression

6. take the integral of the new expression

7. integrate normally\###improper integrals...

...\###case I

f(x) is continuous on [a,∞)

∫f(x) dx\=

lim

∫f(x) dx

b→∞\###case II

f(x) is continuous on (-∞,b]

∫f(x) dx\=

lim

∫f(x) dx

a→-∞\###case III

f(x) is continuous on (-∞,∞)

∫f(x) dx\=

lim lim

∫f(x) dx + ∫f(x) dx

a→-∞ b→∞\###case IV

f(x) is continuous from (a,b]

V.A. @ x\=a

∫f(x) dx\=

lim

∫f(x) dx

q→a⁺\###case V

f(x) is continuous on [a,b)

V.A. @ x\=b

∫f(x) dx\=

lim

∫f(x) dx

k→b⁻\###case VI

f(x) is continuous on [a,c) u (c,b]

V.A. @ x\=c

∫f(x) dx\=

lim lim

∫f(x) dx + ∫f(x) dx

k→b⁻ k→a⁺\###