This article shows that the formula for integration by parts should not be treated as an isolated recipe of elementary calculus, but rather as the integral image of the Leibniz rule for differentiation. The aim is to situate the identity within a broader algebraic architecture, involving derivations, multiplication operators, commutators, formal adjoints, and the abstract structure of integration as a Rota–Baxter-type operator. The present text is deliberately conceptual. Its purpose is to close the circle within the current operational framework by showing that a universal classical identity can be derived from pure algebraic structure, without thereby losing its analytic content.
Ramón Moya (Tue,) studied this question.