complex analysis - Let $f(x) = p(x)/q(x)$ and $deg(p) = deg(q) - 1$. Show that $int_{-infty}^infty f(x) = 0$

I would like to show that $\int_{-\infty}^\infty f(x)dx = 0$ where $f(x) = \dfrac{p(x)}{q(x)}$ and $deg(p) = deg(q) - 1$. Also, $q(x)$ has no real roots. I was considering integrating along the contour $C_R$, where $C_R$ is the real line segment from $-R$ to $R$ and the upper semi circle, in which case

$$\lim_{R \to \infty} \int_{C_R} f(z)dz = \lim_{R \to \infty} \int_{-R}^R f(x) dx+\int_{\Gamma_R}f(z)dz = 2\pi i\sum_{k}Res(f, z_k)$$

where $z_k$ are the zeroes of $q(x)$ in the upper half plane, and $\Gamma_R$ is the upper semicircle. However, I'm not sure where to proceed from here

Any help would be appreciated.