I am attempting to solve the following integral
$\mathrm{Im}\int_{0}^{\infty} \frac{f(x)}{(x-x_0+ia)(x-x_0\pm ib)} dx = ??? $
where $a,b>0$ and then take the $a,b \rightarrow 0$ limit. Also note that $f(x)$ is a real function for every $x \in \mathbb{R}$ and $x_0>0$. I realize that because we are purely interested in the imaginary part of this integral the only nonzero contribution will be very close to the pole $x= x_0 - ia$ or $x=x_0 \mp ib$, Therefore, I think we can integrate this in a small neighborhood with radius $\mathcal{R}$ around $x_0$. Therefore, I think we can write
$ \mathrm{Im}\int_{0}^{\infty} \frac{f(x)}{(x-x_0+ia)(x-x_0\pm ib)} dx = \mathrm{Im}\int_{x_0 - \mathcal{R}}^{x_0 + \mathcal{R}} \frac{f(x)}{(x-x_0+ia)(x-x_0\pm ib)} dx$
I'm trying to solve this by contour integration. I'm thinking we can take the real variable $x$ and promote it to a complex variable $z$ and then integrate across two contours: (1) counterclockwise across a half-circle in the upper complex plane with radius $\mathcal{R}$ and (2) clockwise across a half-circle in the lower complex plane with radius $\mathcal{R}$. Thus, the complex part of the contour integrals will cancel and we are left purely with a real integral. Something like
$ \mathrm{Im}\int_{x_0 - \mathcal{R}}^{x_0 + \mathcal{R}} \frac{f(x)}{(x-x_0+ia)(x-x_0\pm ib)} dx = \frac{1}{2} \mathrm{Im} \left[\oint_{\rm{upper}, |z-x_0| = \mathcal{R} }\frac{f(z)}{(z-x_0+ia)(z-x_0\pm ib)} dz - \oint_{\rm{lower}, |z-x_0| = \mathcal{R} }\frac{f(z)}{(z-x_0+ia)(z-x_0\pm ib)} dz \right] $
where counterclockwise rotations are positive, and hence, we obtain a minus sign for the second term. I was thinking that we could now solve for the residues and take the imaginary part which I found to be
$ \mathrm{Im}\int_{x_0 - \mathcal{R}}^{x_0 + \mathcal{R}} \frac{f(x)}{(x-x_0+ia)(x-x_0\pm ib)} dx = - \pi f'(0)$
when we take the $a,b \rightarrow 0$ limit. This is a technique that a lot of physicists do by taking a real function and introducing a very small imaginary component to it, solve the integral using contour integration and the residue theorem, and then take the limit where the imaginary component goes to zero.
Is there a more elegant way of doing this and is my methodology on the right track, but it has been a while since my complex analysis muscles have been stretched. Any suggestions on how to handle this problem are greatly appreciated!