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Integration by parts

Integration by parts is used when the integrand is a product of two functions, and follows directly from the product rule for differentiation. We choose $u$ and $v'$ so the remaining integral becomes simpler.

\int u\,v'\,dx = uv - \int u'\,v\,dx

integration by parts

Symbols

$u$	differentiated
$v'$	integrated to give v

Example

$\displaystyle\int xe^x\,dx$ with $u=x,\ v'=e^x$ : $= xe^x - \int e^x\,dx = (x-1)e^x + C$ .

Choose

u

so it gets simpler when differentiated (like

x

), and

v'

easy to integrate (like

e^x

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← Definite integral · Substitution →

Part of Calculus: Integration.