Indefinite integral

An integral

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

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 $f$ 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 \ref{*} 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.