On some definite integrals containing the inverse tangent function

Abstract

Evaluations are presented for definite integrals on the form \(\ [ \int_{0}^{\infty} \arctan\left(\frac{2rx}{b{2}+x{2}}\right)f(ax) dx ,]\) for numerous functions \(\ f\) subject to appropriate restrictions on the real numbers \(\ a, r\)  and  \(\ b\). Many new results pertaining to the particular inverse tangent function appearing in the integrand not found in the tables of Gradshteyn and Ryzhik are given.