##  [Integration by Parts](/integration-parts-0) 

 Definition

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

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

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

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

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

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

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

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

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.