Integral for $\frac{x}{x^2+1}cosx$

Question

When computing Fourier transformation I came across these integral:

$$ \int_{\Bbb R}\frac{x \cos x}{1+x^2}\;dx\text{ or } \int_{\Bbb R}\frac{x \sin x}{1+x^2}\;dx $$

Can anyone give me some hints on how to solve them?

Answer 1

Hints: The first one is zero by symmetry. The second one equals $$\Im \int_\mathbb{R} dx\,\frac{x e^{i x}}{1+x^2}$$ which is most easily computed using residues. (Taking the real part of the result also gives back the first integral.)