I'm having troubles evaluating this improper integral

Show: using residues and a closed contour.

So, i used a closed contour C that only contains the singularity . I let R be the radius of the contour and I let R tend to infinity

So,

where

I let

By Integral theorems, I get as and

By the Cauchy Residue Theorem, I get

but I can't get the answer to evaluate. I just wanna know if there is something I'm missing in complex arithmetic, or is it my set-up that is wrong