Module 8 Quick Reference

Techniques of Integration – Calculus I

u-Substitution

∫ f(g(x)) g'(x) dx = ∫ f(u) du,  u = g(x),  du = g'(x) dx

Choose u =

Inner function whose derivative appears in the integrand

Definite integrals

Change bounds: x=a → u=g(a), x=b → u=g(b)

Trig Integrals

Odd power

Strip one, convert rest: sin² = 1 − cos², then u-sub

Both even

sin²x = (1−cos2x)/2, cos²x = (1+cos2x)/2

tan²x = sec²x − 1  |  √(a²−x²): x=a sinθ  |  √(a²+x²): x=a tanθ

Integration by Parts

∫ u dv = uv − ∫ v du
LIATE: Log > Inverse trig > Algebraic > Trig > Exponential

Tabular Method

Derivatives ↓ | Antiderivatives ↓ | Alternate signs: +,−,+,−

Boomerang

If I reappears: solve 2I = expr algebraically

Numerical Integration

Trapezoidal Rule

T = (Δx/2)[f0 + 2f1 + ... + 2fn-1 + fn]

Error: O(1/n²)

Simpson's Rule

S = (Δx/3)[f0 + 4f1 + 2f2 + 4f3 + ... + fn]

Error: O(1/n4), n even

Simpson's is exact for polynomials of degree ≤ 3

Key Results to Memorize

∫ ln x dx = x(ln x − 1) + C  |  ∫ tan x dx = −ln|cos x| + C = ln|sec x| + C
∫ ex sin x dx = (ex/2)(sin x − cos x) + C  |  ∫ ex cos x dx = (ex/2)(sin x + cos x) + C