How do you solve this?
$\displaystyle
\int^{\infty}_{0} x^2 e^{-x^2}
$
What CaptainBlack said but I couldn't resist posting this...
Set $\displaystyle t=x^2$, $\displaystyle x = \sqrt{t}$, dx = $\displaystyle \frac{1}{2}t^{-\tfrac{1}{2}}$.
Integral becomes
$\displaystyle \frac{1}{2} \int_0^{\infty} t^{\tfrac{1}{2}} e^{-t} dt = \tfrac{1}{2}\Gamma(\tfrac{3}{2}) = \tfrac{1}{4}\Gamma(\tfrac{1}{2}) = \tfrac{1}{4}\sqrt{\pi}$
Gamma function - Wikipedia, the free encyclopedia