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


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

The analogue of (1) for an indefinite integral is


The analogue of (1) for a multiple integral is


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 .


[1] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[2] L.D. Kudryavtsev, "Mathematical analysis" , 1–2 , Moscow (1970) (In Russian)
[3] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)


Formula (1) is valid whenever both and are absolutely continuous (cf. Absolute continuity, 3)) on the closed interval . In this generality the integral must be taken in the Lebesgue sense (cf. Lebesgue integral).

For additional references see also Improper integral.


[a1] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)
[a2] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Integration by parts. Encyclopedia of Mathematics. URL:
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