I'd like to Evaluate $$\int\_0^\infty \frac{x\sin x}{1+x^2}$$ The sine function makes the obvious choice $\dfrac{z \sin z}{1+z^2}$ useless since if we integrate over a semicircle sine can become large. I tried $\dfrac{ze^{iz}}{1+z^2}$ but $x\sin x$ is not the real part or imaginary part of $ze^{iz}$. I'm not sure how to choose a related function that will allow me to recover $\dfrac{x\sin x}{1+x^2}.$