# Integration by parts

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One of the methods for calculating integrals. It consists in representing an integral of an expression of the form by an integral of . For a definite integral the formula of integration by parts is (1)

It is applicable under the assumptions that , and their derivatives , are continuous on .

The analogue of (1) for an indefinite integral is (2)

The analogue of (1) for a multiple integral is (3)

Here is a domain in with smooth (or at least piecewise-smooth) boundary ; ; and is the angle between the -axis and the outward normal to . Formula (3) holds if, e.g., , and their first-order partial derivatives are continuous on . If the integrals in (3) are understood as Lebesgue integrals, then the formula is true if and belong to a Sobolev space: , for any with .

How to Cite This Entry:
Integration by parts. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integration_by_parts&oldid=13008
This article was adapted from an original article by V.A. Il'inT.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article