\int _{-\infty }^{\infty }x^2e^{- \frac{x^2}{2}}dx=\int _{-\infty }^{\infty }x \left(x e^{- \frac{x^2}{2}}\right)dx=\left|
\begin{array}{cc}
 u=x & \text{du}=\text{dx} \\
 \text{dv}=x e^{- \frac{x^2}{2}}\text{dx} & v=-e^{-\frac{x^2}{2}}
\end{array}
\right|=-\lim_{x\to \infty } \frac{x}{e^{x^2/2}}+\lim_{x\to  -\infty } \frac{x}{e^{x^2/2}}+\int_{-\infty }^{\infty }e^{- \frac{x^2}{2}} \, dx=0+\sqrt{2 \pi }=\sqrt{2 \pi }