I'm stuck on a question where I'm meant to use Cauchy's integral formula. Here's the question:

Evaluate

along

by Cauchy's integral formula. Computing this integral along the path show that

This is how I approached it:

Let . has poles at and .

The path is a circle centered around with a radius of 1. Consequently, is outside of as . Therefore, according to Cauchy's integral theorem,

where is an arbitrarily small circle around . (I'm not sure if this is right, but I'll show the rest of my work assuming it is.)

Around , is analytic, with

Then, using Cauchy's integral formula,

Therefore,

as .

I've now got no idea how to relate that to the second integral in the question, or if it's even right. Any help is very much appreciated