This is an exercise from Stein & Shakarchi - Complex Analysis

II.7) Prove that for

How do I start?

I know this function has a pole in

I know the function can be rewritten as: or as

What I do not know: what to do? Do I use the residue theorem? Do I look for a nice contour?

(Also in general: how to tackle complex integrals? Step by step.)