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An integral
 
An integral
  
<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/i050/i050550/i0505501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\int f(x)\,dx\tag{*}\label{*}$$
  
of a given function of a single variable defined on some interval. It is the collection of all primitives of the given function on this interval. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i0505502.png" /> is defined on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i0505503.png" /> of the real axis and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i0505504.png" /> is any primitive of it on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i0505505.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i0505506.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i0505507.png" />, then any other primitive of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i0505508.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i0505509.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055011.png" /> is a constant. Consequently, the indefinite integral (*) consists of all functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055012.png" />.
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of a given function of a single variable defined on some interval. It is the collection of all [[primitive function]]s of the given function on this interval. If $f$ is defined on an interval $\Delta$ of the real axis and $F$ is any primitive of it on $\Delta$, that is, $F'(x)=f(x)$ for all $x\in\Delta$, then any other primitive of $f$ on $\Delta$ is of the form $F+C$, where $C$ is a constant. Consequently, the indefinite integral \eqref{*} consists of all functions of the form $F+C$.
  
The indefinite Lebesgue integral of a summable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055013.png" /> is the collection of all functions of the form
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The indefinite Lebesgue integral of a summable function on $[a,b]$ is the collection of all functions of the form
  
<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/i050/i050550/i05055014.png" /></td> </tr></table>
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$$F(x)=\int\limits_a^xf(t)\,dt+C.$$
  
In this case the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055015.png" /> holds, generally speaking, only almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055016.png" />.
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In this case the equality $F'(x)=f(x)$ holds, generally speaking, only almost-everywhere on $[a,b]$.
  
An indefinite Lebesgue integral (in the wide sense) of a summable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055017.png" /> defined on a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055018.png" /> with measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055019.png" /> is the name for the set function
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An indefinite Lebesgue integral (in the wide sense) of a summable function $f$ defined on a measure space $X$ with measure $\mu$ is the name for the set function
  
<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/i050/i050550/i05055020.png" /></td> </tr></table>
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$$\int\limits_Ef(x)\,d_\mu x,$$
  
defined on the collection of all measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050550/i05055022.png" />.
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defined on the collection of all measurable sets $E$ in $X$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)</TD></TR>
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</table>
  
  

Latest revision as of 20:53, 1 January 2019


2010 Mathematics Subject Classification: Primary: 28-XX [MSN][ZBL]

An integral

$$\int f(x)\,dx\tag{*}\label{*}$$

of a given function of a single variable defined on some interval. It is the collection of all primitive functions of the given function on this interval. If $f$ is defined on an interval $\Delta$ of the real axis and $F$ is any primitive of it on $\Delta$, that is, $F'(x)=f(x)$ for all $x\in\Delta$, then any other primitive of $f$ on $\Delta$ is of the form $F+C$, where $C$ is a constant. Consequently, the indefinite integral \eqref{*} consists of all functions of the form $F+C$.

The indefinite Lebesgue integral of a summable function on $[a,b]$ is the collection of all functions of the form

$$F(x)=\int\limits_a^xf(t)\,dt+C.$$

In this case the equality $F'(x)=f(x)$ holds, generally speaking, only almost-everywhere on $[a,b]$.

An indefinite Lebesgue integral (in the wide sense) of a summable function $f$ defined on a measure space $X$ with measure $\mu$ is the name for the set function

$$\int\limits_Ef(x)\,d_\mu x,$$

defined on the collection of all measurable sets $E$ in $X$.

References

[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[2] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[3] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)


Comments

For additional references see Improper integral; Integral.

How to Cite This Entry:
Indefinite integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Indefinite_integral&oldid=11527
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article