# How does integration by parts work?

Sep 12, 2014

Integration by Parts is like the product rule for integration, in fact, it is derived from the product rule for differentiation. It states
$\int u \mathrm{dv} = u v - \int v \mathrm{du}$.

Let us look at the integral
$\int x {e}^{x} \mathrm{dx}$.

Let $u = x$.
By taking the derivative with respect to $x$
$R i g h t a r r o w \frac{\mathrm{du}}{\mathrm{dx}} = 1$
by multiplying by $\mathrm{dx}$,
$R i g h t a r r o w \mathrm{du} = \mathrm{dx}$

Let $\mathrm{dv} = {e}^{x} \mathrm{dx}$.
By dividing by $\mathrm{dx}$
$R i g h t a r r o w \frac{\mathrm{dv}}{\mathrm{dx}} = {e}^{x}$
by integrating,
$R i g h t a r r o w v = {e}^{x}$

Now, by Integration by Parts,
int xe^xdx =xe^x-inte^xdx=xe^x-e^x+C