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When confronted with an integral of the form \begin{equation}  \int_{-\intfy}^{\infty} \int_{-\infty}^{\infty}  e^{-ax^2} \;dx = \sqrt{\frac{\pi}{a}} \end{equation}  There is a standard trick to take the derivative with respect to the parameter $a$ such that \begin{equation}  \frac{d}{da}\int_{-\intfy}^{\infty} \frac{d}{da}\int_{-\infty}^{\infty}  e^{-ax^2} \;dx = \frac{d}{da}\sqrt{\frac{\pi}{a}} \\ \int_{-\intfy}^{\infty} \int_{-\infty}^{\infty}  x^2e^{-ax^2} \;dx = \frac{1}{2}\sqrt{\frac{\pi}{a^3}} \end{equation} However, one could have introduced the parameter $a$ if it wer not there, and set it to one at the end to generate the seemingly arbitrary step.