View source for Indefinite integral

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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 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

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

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====
<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>



====Comments====
For additional references see [[Improper integral|Improper integral]]; [[Integral|Integral]].