Elementary evaluation of $\int_0^\infty{sin^p t/t^q} dt$

Abstract

Let $p$ and $q$ be two positive integers, the goal of this note is to demonstrate, in a very simple and elementary way without using advanced tools, a formula to express the value of the integral $I(p,q)=\int_0^\infty{\sin^p t\over t^q}dt$ when it converges.