The integral is normally computed (e.g. in Ahlfors’ book) to be using complex integration over a suitable almost-rectangular contour. There is also a simple and direct way to get the value of this integral by a substitution and elementary calculus.
First, by the substitution and the identity ,
then using the symmetry of and gives:
whence the result. There are slight technicalities regarding the improperness of these integrals, but they can be directly justified (or one may use the Lebesgue integral).
[Edit (7/25)- Todd Trimble posted solutions to similar integrals, which use the result of this post as a lemma, here. AM]