View source for Integration by parts

One of the methods for calculating integrals. It consists in representing an integral of an expression of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517301.png" /> by an integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517302.png" />. For a [[Definite integral|definite integral]] the formula of integration by parts is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

It is applicable under the assumptions that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517305.png" /> and their derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517307.png" /> are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517308.png" />.

The analogue of (1) for an [[Indefinite integral|indefinite integral]] is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i0517309.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

The analogue of (1) for a [[Multiple integral|multiple integral]] is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173011.png" /> is a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173012.png" /> with smooth (or at least piecewise-smooth) boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173013.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173014.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173015.png" /> is the angle between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173016.png" />-axis and the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173017.png" />. Formula (3) holds if, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173019.png" /> and their first-order partial derivatives are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173020.png" />. If the integrals in (3) are understood as Lebesgue integrals, then the formula is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173022.png" /> belong to a [[Sobolev space|Sobolev space]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173024.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173025.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173026.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,   "Mathematical analysis" , '''1–2''' , Moscow  (1970)  (In Russian)  {{MR|1617334}} {{MR|1070567}} {{MR|1070566}} {{MR|1070565}} {{MR|0866891}} {{MR|0767983}} {{MR|0767982}} {{MR|0628614}} {{MR|0619214}} {{ZBL|1080.00002}} {{ZBL|1080.00001}} {{ZBL|1060.26002}} {{ZBL|0869.00003}} {{ZBL|0696.26002}} {{ZBL|0703.26001}} {{ZBL|0609.00001}} {{ZBL|0632.26001}} {{ZBL|0485.26002}} {{ZBL|0485.26001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)  {{MR|}} {{ZBL|0397.00003}} {{ZBL|0384.00004}} </TD></TR></table>



====Comments====
Formula (1) is valid whenever both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173028.png" /> are absolutely continuous (cf. [[Absolute continuity|Absolute continuity]], 3)) on the closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051730/i05173029.png" />. In this generality the integral must be taken in the Lebesgue sense (cf. [[Lebesgue integral|Lebesgue integral]]).

For additional references see also [[Improper integral|Improper integral]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer  (1965)  {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K.R. Stromberg,   "Introduction to classical real analysis" , Wadsworth  (1981)  {{MR|0604364}} {{ZBL|0454.26001}} </TD></TR></table>