# Integration by parts

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 .

#### References

[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) |

#### Comments

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.

#### References

[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: http://encyclopediaofmath.org/index.php?title=Integration_by_parts&oldid=13008