Integration by parts

$\int u\,dv=uv-\int v\,du$ — reversing the product rule for a product integrand.

Integration by parts reverses the product rule: $\int u\,dv=uv-\int v\,du$ . Given a product integrand, one factor is chosen as $u$ (differentiated) and the rest as $dv$ (integrated) — a good choice makes $\int v\,du$ SIMPLER than the original integral, typically by picking $u$ to be the factor that gets simpler when differentiated (like $x^n$ , which eventually differentiates to a constant).

Examples

$\int x e^x\,dx$ : let $u=x$ , $dv=e^x\,dx$ , so $v=e^x$ , giving $xe^x-\int e^x\,dx=xe^x-e^x+C$ .