New integral formulas inspired by an old integral result

Abstract

In this article, we establish new integral formulas inspired by an old integral result listed in two key books. Some of these
formulas depend on the Euler-Mascheroni constant and extend existing results, broadening their scope and applicability. The
Laplace transforms of certain original functions are emphasized. The proofs are presented in full detail, using rigorous
analytical techniques. These include various geometric and power series expansions, and asymptotic analysis. An open
problem is given at the end.