If I want to do the integrals $I_n$ with residue techniques, what are appropriate functions $f(x)$ to regulate integrals of the form $$ I_n = \int_{-\infty}^\infty \mathrm{d}x \frac{x^n}{(x-a)(x-b)}\times f(x),~~~~a,b\in\mathbb{R}. $$ If I choose a Gaussian $f(x) = e^{-x^2}$, then the contribution from the arc at infinity goes like $$ e^{-R^2\cos{2\phi}}e^{-iR^2\sin{2\phi}}, ~~~~\phi\in(0,\pi) $$ which is not zero as $R\rightarrow \infty$ because of the $e^{-iR^2\sin{2\phi}}$.
So are there any nice functions $f(x)$ that yields zero contribution from the arc at infinity and at the same time make $I_n$ we defined?
\begin{align*} | \mathrm{e}^{\mathrm{i}R^2 \sin 2 \phi} | &= 1 \text{.} \\ \end{align*} The value is confined to a disk. If the radius of the disk shrinks to zero, the phase doesn't matter.