Techniques of Integration

Substitution, integration by parts, partial fractions, and trigonometric substitution — strategies for evaluating integrals that don't yield to direct antidifferentiation.

Choose the integration technique by the structure you see

Inner derivative nearby

∫ 2x cos(x²) dx

u-substitutionu = x²

Product gets simpler if one factor differentiates

∫ x e^x dx

integration by partsu = x, dv = e^xdx

Rational function with factorable denominator

∫ dx / ((x - 1)(x + 2))

partial fractionssplit into simple logs

Square root of a quadratic pattern

∫ sqrt(9 - x²) dx

trig substitutionx = 3 sin theta

The goal is not to decorate the integral. The goal is to turn it into a simpler integral.

Definition

Most integrals cannot be computed by simply reversing a known derivative. Techniques of integration are systematic methods for transforming integrals into ones we can evaluate.

$u$ -substitution (the chain rule in reverse): if the integrand has the form $f(g(x))g'(x)$ , let $u = g(x)$ , so $du = g'(x)\,dx$ :

$\int f(g(x))g'(x)\,dx = \int f(u)\,du$

Integration by parts (the product rule in reverse): $\int u\,dv = uv - \int v\,du$ .

The mnemonic LIATE helps choose $u$ : Logarithms, Inverse trig, Algebraic, Trig, Exponentials — pick $u$ from whichever comes first.

Key properties

Every technique here is a known derivative rule (chain rule, product rule) applied in reverse
Different techniques can apply to the same integral — there's rarely a uniquely "correct" path, just more or less efficient ones
An indefinite integral's antiderivative is only ever determined up to an additive constant $C$
Not every elementary function has an elementary antiderivative (e.g. $e^{-x^2}$ ) — no technique here will find one because none exists

Common mistakes

Forgetting the constant of integration $C$ on indefinite integrals
Picking $u$ poorly in integration by parts: choosing $u$ and $dv$ in the wrong order can make the resulting integral harder, not easier — LIATE is a heuristic, not a guarantee

u-substitution

$\displaystyle\int 2x\cos(x^2)\,dx$

Let $u = x^2$ , $du = 2x\,dx$ .

$\displaystyle\int \cos(u)\,du = \sin(u) + C = \sin(x^2) + C$

Try it

Evaluate $\displaystyle\int x e^{x^2}\,dx$ .

Solution

Let $u = x^2$ , $du = 2x\,dx$ , so $x\,dx = du/2$ .

$\displaystyle\int e^u \frac{du}{2} = \frac{1}{2}e^u + C = \frac{1}{2}e^{x^2} + C$

Related concepts

Needs first

Fundamental Theorem of Calculus Integrals

Applications of Integration

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