Use residues to evaluate the improper integral.
I do not see how to evaluate this integral now. In this section, we use and . I do not see how to use this here. I need hints on how to do this. Thanks in advance.
But what I would do is set up a contour that runs along the real axis from -R to R, then along the semi-circle in the upper complex half-plane back to R.
It should be easy to show that, as R goes to infinity, the the integral around the semi-circle goes to 0 so the integral on the real axis is equal to the residues in the upper complex half-plane. And in that half-plane your integrand is analytic everywhere except where ; in other words at . Find the residue at that point.