Here's another proof:
Start by showing for that Let the integral is equal to find the area of a semicircle with radius hence
Now rearrange this to get what you want.
Oops, I realized that you want to find and I computed Anyway, the link above I gave you contains another proof.
here's my favorite proof of the so-called Gaussian integral:
in the first integral let to get: i.e. is constant.
hence thus call this result (1). on the other hand, from the
definition of it's easily seen that call this (2). now (1) and (2) complete the proof.