Concept preview

What you will be able to do

1. 01State the integration-by-parts formula $\int u\,dv = uv - \int v\,du$ and explain how it is the product rule for derivatives read in reverse.
2. 02Choose $u$ and $dv$ wisely using the LIATE guideline so that the new integral $\int v\,du$ is genuinely simpler, and carry out the method on the classics $\int xe^x\,dx$, $\int x\cos x\,dx$, and $\int \ln x\,dx$.
3. 03Apply integration by parts more than once when needed (repeated parts), and evaluate a definite integral by parts while tracking the crucial minus sign in $-\int v\,du$.

Start from something you already own — the product rule for derivatives. If $u$ and $v$ are functions of $x$, then $\dfrac{d}{dx}\left(uv\right)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}$. Now integrate both sides with respect to $x$. The left side unwinds directly, because integrating a derivative returns the original product: $\int \dfrac{d}{dx}(uv)\,dx = uv$. The right side splits into two integrals: $\int u\,dv + \int v\,du$. Putting the pieces together gives $uv = \int u\,dv + \int v\,du$.