How to prove this relation $\int_0^\infty \frac{\sin(ax)}{x^b}dx\,=\,\frac{1}{2}\pi\,a^{b-1}\frac{1}{\Gamma(b)\sin(\frac{b\pi}{2})}$?
How to integrate $ \int_0^\infty \sin x \cdot x ^{-1/3} dx$ (using Gamma function)