On the Leibnitz Rule for Differentiating Under the Integral Sign

TL;DR

This paper revisits the Leibnitz rule for differentiation under the integral sign, illustrating its application to regular and improper integrals, and showing it often leads to solvable first-order differential equations.

Contribution

It clarifies the conditions and methods for applying Leibnitz's rule, demonstrating its effectiveness in deriving differential equations from integral expressions.

Findings

Applicable to regular and improper integrals

Results in first-order differential equations

Solutions are usually straightforward

Abstract

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.