Novelty of the Laplace-transform approach to the Gaussian integral

Determine whether the approach that evaluates the Gaussian integral ∫_0^∞ e^{-a x^2} dx by applying the Laplace transform with respect to a, interchanging the order of integration to obtain F(s) = ∫_0^∞ 1/(x^2 + s) dx = π/(2√s), and then inverting the transform to recover the integral, is previously known in the mathematical literature.

Background

The paper presents a method to evaluate the Gaussian integral using the Laplace transform: they compute the Laplace transform of f(a) = ∫_0^∞ e^{-a x^2} dx with respect to a, interchange the order of integration to obtain a closed form F(s) = π/(2√s), and then invert the transform to retrieve f(a). The authors note that this approach differs from Laplace's historical method and express uncertainty about whether this particular application is documented in the literature.

Clarifying the novelty of this technique has bibliographic importance and could inform whether the method should be cited as new or as a rediscovery of an existing approach.