Evaluating, $\int_0^{\infty} \frac{x\sin(x)}{x^2 + 1}dx$

Question

Evaluate,

$\int_0^{\infty} \frac{x\sin(x)}{x^2 + 1}dx$

using methods from complex analysis.

Is there a theorem related to this problem that someone can suggest and/or give a hint that would be helpful in evaluating this integral?

Answer 1

Using partial fraction decomposition $$\frac x{x^2+1}=\frac{1}{2 (x+i)}+\frac{1}{2 (x-i)}$$ Now $$\int \frac {\sin(x)}{x+a}\,dx=\cos (a) \text{Si}(a+x)-\sin (a) \text{Ci}(a+x)$$ $$\int_0^\infty \frac {\sin(x)}{x+a}\,dx=\text{Ci}(a) \sin (a)-\text{Si}(a) \cos (a)+\frac{1}{2} \pi \cos (a)$$

Apply it, simplify and admire the beauty of the result.