New identities for the Laplace transform and their applications

Abstract

In this paper, we begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solution to the Pythagorean equation $a^2 + b^2 = c^2$, these closed forms become even more intriguing. This approach allows us to provide new integral representations for the error function. Additionally, by leveraging an identity we derived for the inverse Laplace transform and applying a result based on Srivastava and Yürekli's identity, we provide a closed form for a nontrivial generalized integral.