Module 8: Techniques of Integration
Calculus I – Study Guide
Learn Without Walls
1. u-Substitution
∫ f(g(x)) g'(x) dx = ∫ f(u) du where u = g(x)
Choose u = inner function. Its derivative must appear (up to a constant) in the integrand.
Common Patterns
| Integral | Let u = |
|---|---|
| ∫ f(ax + b) dx | ax + b |
| ∫ xn−1 f(xn) dx | xn |
| ∫ f(sin x) cos x dx | sin x |
| ∫ f(ln x)/x dx | ln x |
Definite integrals: Change bounds to u-values. u(a) becomes the lower limit, u(b) the upper limit.
2. Trig Integrals
∫ sinmx cosnx dx
Odd power: Strip one factor, convert the rest using sin² + cos² = 1, then u-sub.
Both even: Use power-reduction: sin²x = (1 − cos 2x)/2, cos²x = (1 + cos 2x)/2.
tan²x = sec²x − 1 (useful for tangent integrals)
Trig Substitution
| Expression | Substitution |
|---|---|
| √(a² − x²) | x = a sin θ |
| √(a² + x²) | x = a tan θ |
| √(x² − a²) | x = a sec θ |
3. Integration by Parts
∫ u dv = uv − ∫ v du
LIATE rule: Logarithmic > Inverse trig > Algebraic > Trig > Exponential. Pick u from what appears first.
Tabular method: For polynomial × exponential/trig, list derivatives and antiderivatives in columns, alternate signs, multiply diagonally.
Boomerang: If the original integral reappears after two rounds of IBP, solve algebraically: 2I = (expression), so I = (expression)/2.
Key results: ∫ ln x dx = x(ln x − 1) + C. ∫ ex sin x dx = (ex/2)(sin x − cos x) + C.
4. Numerical Integration
Trapezoidal Rule
Tn = (Δx/2)[f(x0) + 2f(x1) + ... + 2f(xn−1) + f(xn)]
Error: |ET| ≤ M(b − a)³/(12n²), where M = max|f''|.
Simpson's Rule
Sn = (Δx/3)[f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + f(xn)]
Requires even n. Error: |ES| ≤ K(b − a)5/(180n4), where K = max|f(4)|. Exact for polynomials of degree ≤ 3.