# Indefinite integral

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 $X$.

How to Cite This Entry:
Indefinite integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Indefinite_integral&oldid=43640
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article