Assume that $|f(z) / z| \rightarrow 0$ as $|z| \rightarrow \infty$ and that $f(z)$ is analytic in the upper half-plane (including the real axis). Evaluate

$$\mathcal{P} \int_{-\infty}^{\infty} \frac{f(x)}{x\left(x^{2}+a^{2}\right)} d x$$

where $a$ is a positive real number.

[You must state clearly any standard results involving contour integrals that you use.]