Calculate: $$\int_{0}^{\infty}\frac{\sin x}{x^{3}+x}\mathrm{d}x$$
My try:
Let's split to a pacman style path, with little circle around the singularity and 2 rays from $0$ to $\infty$:
$\displaystyle \int_{0}^{\infty}\frac{\sin x}{x^{3}+x}\mathrm{d}x=\Im\int_{|z|=\varepsilon}\frac{\varepsilon e^{\theta i}}{(\varepsilon e^{\theta i})^{3}+\varepsilon e^{\theta i}}\mathrm{d}\theta+\Im\int_{0}^{\infty}\frac{re^{\delta i}}{(re^{\delta i})^{3}+re^{\delta i}}\mathrm{d}r+\Im\int_{0}^{\infty}\frac{re^{-\delta i}}{(re^{-\delta i})^{3}+re^{\delta i}}\mathrm{d}r$
where $\varepsilon \to 0$, and then the integral is $1$ therefore it's imaginary part is $0$.
$\delta$ is again an angle as small as we want (as it need not contain the pole at $\pm i$. According to the residue theorem the domain contains no poles therefore their sum must be zero, therefore the whole integral must be $0$, but this is not the correct answer. What claim I make is wrong?