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Substitution

Substitution is the integration counterpart of the chain rule. By replacing part of the integrand with a new variable $u$ , a difficult integral can become a simple one.

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

the substitution rule

Symbols

u

new variable,

u=g(x)

Example

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

Look for a factor in the integrand that is the derivative of the rest - that is often the sign of the right substitution.

Practise integration for free →

← Integration by parts · Limits and L'Hôpital →

Part of Calculus: Integration.