I am trying to work through these simple exercises but it invariably blends together and I get lost. Here is one of them:

Use the Complex FTC and/or Cauchy-Goursat Theorem to show that

, where

is the circle with centre 0 and radius 1, oriented as you prefer, and when:

1.

I'm not sure how to proceed. I know that

,

Do I proceed like:

or

Do I just integrate the original equation and forget about polar coordinates?

Surely this is a simple question. Should I be doing something else? Is there any way to avoid tedious integration by parts?