# How do you know to use integration by parts?

Sep 26, 2014

Since the integration by parts is essentially the product rule for integration, an integral whose integrand is the product of functions is most likely to call for the integration by parts.

Let us look at a simple example.

$\int x {e}^{x} \mathrm{dx}$

As you can see, the integrand is the product of $x$ and ${e}^{x}$, so the integration by parts is a good one to try.

Let $u = x$ and $\mathrm{dv} = {e}^{x} \mathrm{dx}$,
$R i g h t a r r o w \mathrm{du} = \mathrm{dx}$ and $v = {e}^{x}$

By Integration by Parts: $\int u \mathrm{dv} = u v - \int v \mathrm{du}$,

$= x {e}^{x} - \int {e}^{x} \mathrm{dx}$

By Exponential Rule,

$= x {e}^{x} - {e}^{x} + C$