\int_{-\infty}^{+\infty}x^2e^{-\alpha x^2}dx=\frac{1}{2\alpha}\sqrt{\frac{\pi}{a}}
\int_{-\infty}^{+\infty}x^2e^{-\alpha^2 x^2}dx=\frac{\sqrt \pi}{2\alpha^3}
\int_{-\infty}^{+\infty}xe^{-\alpha x^2}dx=0
\int_{-\infty}^{+\infty}e^{-\alpha^2 x^2}dx=\frac{\sqrt \pi}{\alpha}
\int_{-\infty}^{+\infty}e^{-\alpha x^2}dx=\sqrt{\frac{\pi}{\alpha}}
\int_{-\infty}^{+\infty}e^{-(\alpha x^2+bx)}dx=\sqrt{\frac{\pi}{\alpha}}e^{\frac{b^2}{4a}}