An Integral with Three Parameters

Abstract

In this paper we give an exact expression for the integral \begin{eqnarray} I(a, b; r) = \int_{0}^{\infty} \left[ \frac{x^{2}}{x^{4} + 2a x^{2} + 1} \right]^{r} \cdot \frac{x^{2} + 1}{x^{b} + 1} \cdot \frac{dx}{x^{2}} \nonumber \end{eqnarray} in terms of Euler's beta function. Several classical integrals are deduced from it. The expression for $I(a,b;r)$ provides a method for unifying a large class of integrals.