Integration Techniques

While basic integration rules work for simple functions, most real-world integrals require specialized techniques. Master these methods to solve any integral.

1. U-Substitution (The Chain Rule in Reverse)

U-substitution is the most commonly used integration technique. It's essentially the reverse of the chain rule for derivatives.

The Substitution Formula

\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du = F(u) + C = F(g(x)) + C

Where u = g(x) and du = g'(x)dx

Steps for U-Substitution

Identify an "inner function" g(x) to substitute
Let u = g(x) and find du = g'(x)dx
Rewrite the entire integral in terms of u
Integrate with respect to u
Substitute back to get the answer in terms of x

Example

Evaluate: $\int 2x \cdot cos(x²) dx$

Solution:

Let u = x², then du = 2x dx
∫ cos(u) du = sin(u) + C
Answer: sin(x²) + C

2. Integration by Parts

Integration by parts is the reverse of the product rule. Use it when you have a product of functions that can't be solved by substitution.

Integration by Parts Formula

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

LIATE Rule for Choosing u

Choose u in this priority order (set dv to be the rest):

Logarithmic functions (ln x, log x)
Inverse trig functions (arcsin, arctan)
Algebraic functions (x², x, polynomials)
Trigonometric functions (sin, cos, tan)
Exponential functions (eˣ, aˣ)

Example

Evaluate: $\int x \cdot eˣ dx$

Solution:

Let u = x (algebraic), dv = eˣ dx
Then du = dx, v = eˣ
∫ x·eˣ dx = x·eˣ - ∫ eˣ dx = x·eˣ - eˣ + C
Answer: eˣ(x - 1) + C

3. Partial Fractions

Use partial fractions to integrate rational functions (fractions with polynomials).

Key Idea

Break a complex fraction into simpler fractions that are easier to integrate.

\frac{P(x)}{Q(x)} = \frac{A}{(x-a)} + \frac{B}{(x-b)} + \cdots

Cases for Partial Fractions

Case 1: Distinct Linear Factors

\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

Case 2: Repeated Linear Factors

\frac{1}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}

Case 3: Irreducible Quadratic Factors

\frac{1}{x^2+a^2} = \frac{Ax+B}{x^2+a^2}

Example

Evaluate: $\int 1/[(x-1)(x+1)] dx$

Solution:

Decompose: 1/[(x-1)(x+1)] = A/(x-1) + B/(x+1)
Solving: A = 1/2, B = -1/2
∫ [1/2·1/(x-1) - 1/2·1/(x+1)] dx
Answer: (1/2)ln|x-1| - (1/2)ln|x+1| + C = (1/2)ln|(x-1)/(x+1)| + C

4. Trigonometric Substitution

Use trigonometric substitution for integrals containing √(a² - x²), √(a² + x²), or √(x² - a²).

For √(a² - x²)

x = a\sin\theta

Uses: 1 - sin²θ = cos²θ

For √(a² + x²)

x = a\tan\theta

Uses: 1 + tan²θ = sec²θ

For √(x² - a²)

x = a\sec\theta

Uses: sec²θ - 1 = tan²θ

5. Trigonometric Integrals

Special strategies for integrals involving powers of trig functions.

∫ sinᵐx cosⁿx dx

If m is odd: Save one sin x, convert rest to cos using sin²x = 1 - cos²x, use u = cos x
If n is odd: Save one cos x, convert rest to sin using cos²x = 1 - sin²x, use u = sin x
If both even: Use half-angle identities

Half-Angle Identities

\sin^2 x = \frac{1 - \cos 2x}{2}

\cos^2 x = \frac{1 + \cos 2x}{2}

6. Reduction Formulas

Reduction formulas reduce the power of an integrand, letting you solve complex integrals step by step.

Power of Sine

\int \sin^n x \, dx = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n}\int \sin^{n-2}x \, dx

Power of Cosine

\int \cos^n x \, dx = \frac{\cos^{n-1}x \sin x}{n} + \frac{n-1}{n}\int \cos^{n-2}x \, dx

How to Choose the Right Technique

Check for basic formulas first — Can you integrate directly?
Look for substitution opportunities — Is there an inner function and its derivative?
Products of different types? — Use integration by parts (LIATE)
Rational function? — Use partial fractions
Square roots of quadratics? — Use trig substitution
Powers of trig functions? — Use trig identities or reduction formulas

Put These Techniques to Practice

Practice Problems Review Rules