Lesson 6 of 14

Integration by Parts

The product rule in reverse, the LIATE heuristic, and the tabular shortcut.

The formula

Integrate the product rule $(uv)' = u'v + uv'$ and rearrange:

\int u\,dv = uv - \int v\,du

You trade one integral for another, which pays off when $\int v\,du$ is simpler. Example: $\int x e^x dx$ . Take $u = x$ , $dv = e^x dx$ , so $du = dx$ , $v = e^x$ :

\int x e^x dx = x e^x - \int e^x dx = (x-1)e^x + C

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