We begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. Using the solution to the Pythagorean equation \(a^2 + b^2 = c^2\), we find that these closed forms become even more intriguing. This approach enables us to propose new integral representations for the error function, with some of the resulting formulas appearing as Fourier sine and cosine transforms.