On the Leibnitz Rule for Differentiating Under the Integral Sign

This Note revisits the Leibnitz integral calculus method based on differentiation under the integral sign with respect to a parameter either already existing or introduced ad hoc. Through several cases exemplifying the method, it is shown that this approach, applicable to regular and, under certain conditions, to improper integrals as well, results in a 1st order differential equation whose solution is usually straightforward.