$\displaystyle
\int^{\infty}_{0}x^{2}e^{-x^2/2}dx$
$\displaystyle \int^{\infty}_{0}x^{2}e^{-x^2/2}dx$
$\displaystyle =\int^{\infty}_{0}x.xe^{-x^2/2}dx$
$\displaystyle =[-xe^{-x^{2}/2}]_{0}^{\infty}+\int^{\infty}_{0}e^{-x^2/2}dx$
$\displaystyle =0+\int^{\infty}_{0}e^{-x^2}\sqrt{2}dx$
$\displaystyle =\sqrt{\frac{\pi}{2}}$