The question is in the title. How should one go about solving the above integral using real analysis methods? The standard "trick" of squaring the integral and converting it into an integral over a region in polar coordinates doesn't seem to work here, because the corresponding 2 dimensional region (a square) in the $xy$ plane does not have a simple description in polar coordinates. Wolfram Alpha gives a solution in terms of $\operatorname{erfc(x)}$, but I am wondering if there is an actual closed form for this integral not in terms of special functions.
Thanks!
