Your integral can be written as follows:
You can double-click the integral to see the LaTeX code that generated it.
I agree that you can't integrate directly over a pole. What you're going to have to do is skirt the pole by doing an semicircle around the pole. It's a first-order pole, by the way, since the numerator has a first-order zero at zero. Once you've computed the integral over the semicircle, you can add to that the other two integrals from and Make sense?