Lobachevsky Integral Formula

Theorem
Let $f$ be a continuous function, periodic in $\pi$. Then,


 * $\displaystyle \int_0^\infty \frac{\sin x} x f\left({x}\right) \, \mathrm d x = \int_0^{\frac \pi 2} f\left({x}\right) \, \mathrm d x$