U-substitution

Reversing the chain rule to integrate a composite function, by renaming its inner piece $u$ .

U-substitution integrates a composite integrand by identifying an inner function $u=g(x)$ whose derivative $du=g'(x)\,dx$ also appears (up to a constant) elsewhere in the integrand, then rewriting the entire integral in terms of $u$ alone. For a DEFINITE integral, the bounds must be converted too — from values of $x$ to the corresponding values of $u$ — so the antiderivative never needs to be converted back to $x$ .

Examples

$\int 2x(x^2+1)^5\,dx$ with $u=x^2+1$ , $du=2x\,dx$ becomes $\int u^5\,du=\frac{u^6}{6}+C=\frac{(x^2+1)^6}{6}+C$ .