I have some trouble evaluating the following integral using complex analysis:

I believe the trick is to complexify the integrand, let and then make a contour:

And . Then we can evaluate with the Cauchy-integral formula.

By Jordan's Lemma . We like to evaluate

Can someone tell me how far off I am, and maybe the correct approach for this. My problem is, I have no idea how to evaluate , the partial residu around the second order singularity z=0.

So is this the right approach, if not, what is? And if yes, how to evaluate the afore mentioned partial residue.