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\end{equation}  for $a>0$.  There is a standard trick to take the derivative with respect to the parameter $a$ such that \begin{equation} \frac{d}{da}\int_{-\infty}^{\infty} e^{-ax^2} \;dx = \frac{d}{da}\sqrt{\frac{\pi}{a}} \\ 
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\end{equation}  for $a>0$.  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.