How to calculate this principal value? The numeric answer is amazing to me.

Question

I need to calculate the following principal value: $$ PV\int_{\pi/a}^\infty \frac{x}{x^2-x_0^2} cos(\sqrt{x^2-(\pi/a)^2}r)\frac{\sqrt{x_0^2-(\pi/a)^2}}{\pi\sqrt{x^2-(\pi/a)^2}}dx $$ At first I thought there was no analytical answer to it, but when I did the numerical calculation, it turns out that the answer is highly identical to $$ -sin(\sqrt{x_0^2-(\pi/a)^2}r)/2 $$ Can anyone tell me how to analytically prove this result? Thank you!