The identity transforming the integral of a product of functions into a boundary term minus the integral of one function times the derivative of the other: ∫ u dv = uv − ∫ v du, valid under appropriate differentiability and integrability conditions.

Principle

It is the integral form of the product rule for differentiation: since (uv)' = u'v + uv', integrating both sides yields the integration by parts identity; the method trades differentiation of one factor for integration of the other.

Demonstration

To compute ∫ ln(x) dx on (1,a), set u = ln(x), dv = dx. Then du = dx/x and v = x, giving ∫ ln(x) dx = x ln(x) − ∫ x · (1/x) dx = x ln(x) − x + C.

Misapplication

Applying integration by parts without ensuring that the boundary term uv is finite or that v and du are integrable; repeated application without improvement of integrability can produce divergent or ill-defined expressions.

Consequence

When correctly applied, integration by parts simplifies integrals, yields reductions for special functions, provides weak formulations of differential equations, and underpins definitions of distributional derivatives and Sobolev spaces.

Reversal

The reverse viewpoint is deriving the product rule from the integral identity, or viewing IBP as integrating the derivative of a product; failing to account for endpoints reverses the equality by introducing missing boundary contributions.

Boundary

Requires that u be absolutely continuous (or differentiable with integrable derivative) and v an antiderivative of dv with appropriate integrability so that uv and ∫ v du are defined; does not apply naïvely to purely distributional factors without reinterpretation.

Semantic Tension

Tension exists between integration by parts as a formal algebraic manipulation and its rigorous use in spaces of weak derivatives: in classical calculus endpoints and integrability are explicit, while in Sobolev theory IBP is recast as duality.

Synthesis

Integration by parts converts a product integral into boundary terms plus a simpler integral by using the product rule; its rigorous application depends on endpoint behavior and integrability and it bridges elementary calculus with weak derivative frameworks.