Integration by Parts

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When we looked at integrating the natural logarithm, we used a little trick: we took the product rule

$\displaystyle (uv)' = u'v + uv'$ ,

integrated it to get

$\displaystyle uv = \int v\,du + \int u\,dv$ ,

and made a clever choice of $u$ and $v$ to find the integral we wanted. This is called integration by parts, and it’s usually written like this:

$\displaystyle\int u\,dv = uv - \int v\,du$ .