The question is:

Evaluate 
$\displaystyle
\
\int\_{- \infty}^{\infty} \frac{x e^{2ix}}{x^2 - 1}\,dx
\
$
using the contour below. 

(Explain what happens on each part of the contour.)

![enter image description here](https://i.stack.imgur.com/JLBDr.jpg)

First of all, isn't this a bad choice of contour? (Since it branch cuts between the 2 singularities)

If we have to do it this way. Do we have to do it in 6 parts:

The upper curve CR

The lower left line segment

The lower right line segment

The center line segment

The 2 arcs around the singularities: Cr1 and Cr2

This seems to be a very complicated situation. How do I do each of these steps?

Is it residue theorem I have to use? How do I apply it in this situation?

I have spent an hour reading about this but didn't get it. I have very limited time to burn on this specific kind of problem. So helps are appreciated, either an answer with steps or intuitive hints are appreciated.

Thanks.