\int _1^{\infty }x^2e^{- \frac{x^2}{2}}dx=\int _1^{\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^{\frac{x^2}{2}}}+e^{-\frac{1}{2}}+\int_1^{\infty } e^{- \frac{x^2}{2}} \, dx=\left|\text{Smena} \frac{x}{\sqrt{2}}\right.=t|=\frac{1}{\sqrt{e}}+\sqrt{2}\int_{\frac{1}{\sqrt{2}}}^{\infty } e^{-t^2} \, dt