# Difference between revisions of "Indefinite integral"

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An integral | An integral | ||

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

− | of a given function of a single variable defined on some interval. It is the collection of all | + | 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 $[a,b]$ is the collection of all functions of the form | The indefinite Lebesgue integral of a summable function on $[a,b]$ is the collection of all functions of the form | ||

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

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

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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 | 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 | ||

− | $$\int\limits_Ef(x)d_\mu x,$$ | + | $$\int\limits_Ef(x)\,d_\mu x,$$ |

defined on the collection of all measurable sets $E$ in $X$. | 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> | + | <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> | ||

## 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=32817