Integration by parts

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Integration by Parts (Partial integration)

Integration by parts (or partial integration) is one of the most useful basic calculus operations. It is a generalization of the fundamental theorem of calculus for the case of the product of two functions.

Fundamental theorem of Calculus

The classical definition of the definite integral of a function on the closed interval as [1]

Integration by parts

The second important result from calculus is the form of the Fundamental Theorem known as integration by parts or partial integration. If we consider two differentiable functions , then we may write by the product rule

Integrating both sides from to yields

which may be written more simply as

Recognizing that the term on the left is an exact differential, we may write

This expression may be rearranged to yield the familiar form of integration by parts

A generalization of integration by parts to higher dimensions is the Divergence Theorem also known as Gauss's Law, particularly when applied in electromagnetism.


  1. Greenspan, Harvey Philip, and David J. Benney. Calculus: an introduction to applied mathematics. H, P. GREENSPAN, 1997.