# How can you integrate xe^(x^2) dx using integration by parts?

May 19, 2015

We do not need to integrate by parts, but since that is what is specified:

We can integrate $x {e}^{{x}^{2}} \mathrm{dx}$ by sustitution $w = {x}^{2}$ and we end up with

$\int x {e}^{{x}^{2}} \mathrm{dx} = \frac{1}{2} {e}^{{x}^{2}} + C$

Therefore, to use parts, I will choose $u = 1$ and $\mathrm{dv} = x {e}^{{x}^{2}} \mathrm{dx}$

This makes $\mathrm{du} = 0 \mathrm{dx}$ and $v = \frac{1}{2} {e}^{{x}^{2}}$

The parts formula gives us;

$\frac{1}{2} {e}^{{x}^{2}} - \int 0 \mathrm{dx}$

$= \frac{1}{2} {e}^{{x}^{2}} + C$