Contour integration with absolute value

by JohnnyB   Last Updated January 16, 2018 10:20 AM

I am trying to use contour integration to evaluate the integral $$ \int_\mathbb{R}\frac{|x|e^{i\omega x}}{x^2+ a}dx \,, $$ where $a>0$ and $\omega$ can take any real value. I am worried about the existence of the absolute value and I am not sure how to proceed.

Any help appreciated.

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