I need to prove the following:

$\displaystyle \int^\infty_0 x^2e^{-x^2}=\frac{1}{2}\int^\infty_{0} e^{-x^2}$

Im having trouble integrating this, although I do know, based on the Gamma function, that $\displaystyle \frac{1}{2}\int^\infty_{0} e^{-x^2}=\frac{\sqrt{\pi}}{4}$

So really what I need to show is that $\displaystyle \lim_{b\to \infty}\int^b_0 x^2e^{-x^2}=\frac{\sqrt{\pi}}{4}$

But when I try to integrate by parts, Im getting even more confusing terms, can someone help me out?