\int_{-\infty}^{\infty}x^2 e^{-ax^2}dx=-\int_{-\infty}^{\infty}\frac{d}{da}e^{-ax^2}dx=-\frac{d}{da}\int_{-\infty}^{\infty}e^{-ax^2}dx=-\frac{d}{da}\sqrt{\frac{\pi}{a}}=\frac{1}{2}\sqrt{\frac{\pi}{a^2}}