  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 method allows us to
provide new integral representations for the error function. Following this, we
use the Fourier transform to derive formulas for the Glasser and Widder
potential transforms, leading to several new and interesting corollaries. As
part of the applications, we demonstrate the use of one of these integral
formulas to provide a new real analytic proof of Euler's reflection formula for
the gamma function. Of particular interest is a generalized integral involving
the Riemann zeta function, which we also present as an application.
