A unique method to evaluate the general integral $\int_0^\infty dx\frac{\sin ^a px \cos ^c qx}{x^b} $

Abstract

All integrals available in literature and books, that are related to Sinc(=sin x/x) function, are special cases of the general form of the integral given in the title. The evaluation of the integral is divided into two cases (i) $a$ and $b$ of same parity, which is easier to evaluate and (ii) $a$ and $b$ of different parity, a difficult case. Amazingly and may be for the first time, a divergent integral is used in evaluating this difficult case with the help of a simple but a special combinatorial expression. The combinatorial identity is derived from the power reduction formula of the Sines and Cosines. The method adopted in this paper is unique and makes it relatively easy to evaluate this integral.