An Ahmed-like integral

TL;DR

This paper extends a probabilistic approach to evaluate a class of integrals related to the Ahmed integral, deriving new identities and demonstrating the systematic generation of Ahmed-type integrals from higher powers of Gaussian integrals.

Contribution

It generalizes Pla's method to higher powers of Gaussian integrals, producing new Ahmed-type integral identities and illustrating a systematic approach for their derivation.

Findings

Derived a new integral identity involving arctangent and Gaussian integrals.

Extended Pla's probabilistic method to the fifth power of Gaussian integrals.

Showed that this approach can generate a family of Ahmed-type integrals.

Abstract

The so-called Ahmed integral $$ \int_{0}^{1}\frac{\arctan\left(\sqrt{2+x^{2}}\right)}{(1+x^{2})\sqrt{2+x^{2}}}\,\mathrm{d} x=\frac{5\pi^{2}}{96}, $$ has attracted considerable interest since its appearance in the "American Mathematical Monthly" in 2001. Several proofs and extensions have been proposed, including a probabilistic multivariate approach introduced by Pla based on powers of the Gaussian integral. In this note, we extend Pla's method to the fifth power of the Gaussian integral. By expressing this power as a sequence of iterated integrals and performing successive reductions, we obtain a new integral identity closely related to Ahmed's integral. In particular, we prove that $$ \int_{0}^{1}\frac{\arctan\!\left(\sqrt{\displaystyle\frac{2+x^2}{4+x^2}}\right)}{(1+x^{2})\sqrt{2+x^{2}}}\,\mathrm{d} x=\frac{\pi^2}{30}. $$ The derivation suggests that Pla's technique can systematically generate a family of Ahmed-type integrals associated with higher powers of the Gaussian integral.