Evaluate $\int_{0}^{\infty} \frac{\sin x-x\cos x}{x^2+\sin^2x } dx$ The integral  $$\int_{0}^{\infty} \frac{\sin x-x\cos x}{x^2+\sin^2x } dx$$ admits
a nice closed form. The question is: How to evaluate it by hand.

$$ I=\int_{0}^{\infty} \frac{\sin x-x \cos x}{x^2+\sin^2 x}  dx= - \int_{0}^{\infty}\frac{\frac {x\cos x -\sin x}{x^2}}{1+(\frac{\sin x}{x})^2}dx=
-\int_{1}^{0} \frac{dt}{1+t^2}=\frac{\pi}{4}.$$
