Calculus - Techniques of Integration

Techniques of Integration

Integration by Parts

$\int u \, dv = uv - \int v \, du$

LIATE rule: Order for choosing u Logarithmic → Inverse trig → Algebraic → Trigonometric → Exponential

Trigonometric Integrals

Powers of sin x and cos x:

• If odd power, factor out one term
• If even powers, use identities: sin²x = (1-cos 2x)/2, cos²x = (1+cos 2x)/2

Powers of tan x and sec x:

• ∫ sec^(2n-1)x: integration by parts with u = sec^(n-1)x
• ∫ tan^m x sec^n x: special cases

Trigonometric Substitution

• For √(a²-x²): use x = a sin θ
• For √(a²+x²): use x = a tan θ
• For √(x²-a²): use x = a sec θ

Check: dx substitutions and back-substitution to original variable.

Partial Fractions

Decompose rational functions:

Distinct linear factors: A/(x-a) + B/(x-b) + …

Repeated linear factors: A/(x-a) + B/(x-a)² + …

Quadratic factors: (Ax+B)/(x²+px+q) for irreducible quadratics