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

TL;DR

This paper introduces a novel method for evaluating a broad class of integrals involving sine and cosine functions divided by a power of x, including divergent cases, using combinatorial identities derived from power reduction formulas.

Contribution

It presents a unique approach that simplifies the evaluation of complex integrals related to the Sinc function, including handling divergent cases with combinatorial techniques.

Findings

Unified evaluation method for integrals involving sine and cosine functions

Inclusion of divergent integrals in the evaluation process

Derivation of combinatorial identities from power reduction formulas

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.