New identities for the Laplace transform and their applications

In this paper, we apply the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. By utilizing the solutions to the Pythagorean equation $ a<^>2 + b<^>2 = c<^>2 $ a2+b2=c2, these closed forms become even more intriguing. This approach allows us to provide new integral representations for the error function. Furthermore, by leveraging an identity we derived for the inverse Laplace transform and applying a result based on Srivastava and Y & uuml;rekli's identity, we provide a closed form for a nontrivial generalized integral.