The integral is $\displaystyle I=\int _0 ^{+\infty} \frac{x \sin (x)}{x^2+2}dx$. I'll probably choose a semi circle as contour since the poles aren't on the real axis. Furthermore I think that $\displaystyle I=\frac{1}{2} \int _{-\infty}^{+\infty } \frac{x \sin (x)}{x^2+2}dx$.

Let $\displaystyle f(z)=\frac{z \sin z}{z^2+2}$. The singularities of f are when $\displaystyle z=\pm i \sqrt 2$.

Now I want to calculate the residue of f at say one singularity. But it gives infinity I think. So I believe it's a pole of order greater than 1 but I'm unsure. I tried to use the Taylor series of sin(z) in order to see what happens in $\displaystyle z=i \sqrt 2$ but I didn't reach anything.

So I don't know how to go further.