Evaluating an integral of a real function using the residue theorem

Question

Using the residue theorem, I have shown that $\oint_C f(z) dz = \frac{\pi i}{e^2}$ where $f(z) = \frac{ze^{iz}}{(z^2+4)^2}$ and $C$ is the closed curve consisting of the horizontal line $y = 0$ from $x = -R$ to $x = R$ and the semicircle $y = \sqrt{R^2-x^2}$. How can I use this result to evaluate $\int_{-\infty}^{\infty} \frac{x \sin(x)}{(x^2+4)^2}$? I know that it involves using the fact that $\Im{(e^{ix})} = \sin(x)$ but I have no idea exactly how to use it.