nLab indefinite integral

Idea

An indefinite integral is something less definite than a definite integral. Whereas a definite integral is typically some kind of number or other concrete quantity, an indefinite integral is typically another variable quantity of the same type as the integrand.

The term ‘indefinite integral’ is itself rather indefinite, having been used for a variety of slightly different concepts. Both semidefinite integrals and antiderivatives are more precise versions of indefinite integrals. The fundamental theorem of calculus is basically the theorem that these different kinds of indefinite integral are essentially the same thing.

Definitions and notation

To begin with, we will discuss the integration of real-valued functions on the real line, but much of this can be generalized to other contexts. So let $f$ be a partial function from $\mathbb{R}$ to $\mathbb{R}$; typically, the domain of $f$ will be an interval, but we do not require this.

Definition

If $a$ is a real number (usually in the domain of $f$, or at least in the domain's closure), then the semidefinite integral of $f$ from $a$ (or with initial point $a$) is the function

$$x \mapsto \int_a^x f(t) \,\mathrm{d}t .$$

(If $x \lt a$, then we must define $\int_a^x$ as $-\int_x^a$.)

The semidefinite integral is defined in terms of the definite integral. We can put names such as ‘Riemann’ and ‘Lebesgue’ between ‘semidefinite’ and ‘integral’ to specify a particular kind of definite integral to be used. Note that the domain of the semidefinite integral is an interval containing $a$ and contained in the domain of $f$ (or at least in its closure if we allow improper integral?s or integrating almost functions). If we start by defining $f$ as a locally integrable function? on a closed interval $I$ containing $a$, then the semidefinite integral will also have $I$ as its domain.

The value at $x$ of the semidefinite integral from $a$ may be denoted

$$\int_a f(x) \,\mathrm{d}x$$

for short. Notice that this notation has no dummy variable; we only need to introduce the dummy variable $t$ to unfold the definition. (Indeed, some writers will abuse notation, writing $\int_a^x f(x) \,\mathrm{d}x$ for the semidefinite integral.) But as in $\mathrm{d}y/\mathrm{d}x$, the $x$ here is not a free variable either, since we cannot freely use substitution; it has to be viewed as variable quantity? instead. Rather, if you want to evaluate $\int_a f(x) \,\mathrm{d}x$ when $x$ is some number $b$, then the notation for this is $(\int_a f(x) \,\mathrm{d}x)|_{x=b}$ following the general logic of variable quantities; but unwrapping the definition of semidefinite integral, this can also be written as $\int_a^b f(x) \,\mathrm{d}x$. (In each of these, $x$ has now become a dummy variable.)

Definition

If $a$ and $C$ are real numbers (with $a$ in the domain of $f$ or its closure), then the indefinite integral of $f$ from $a$ with initial value $C$ is the function

$$x \mapsto C + \int_a^x f(t) \,\mathrm{d}t .$$

We may write this value as $C + \int_a f(x) \,\mathrm{d}x$ for short.

This is only one of the meanings of ‘indefinite integral’, but it is the only one that doesn't have alternative unambiguous terminology. Note that $C$ is the value of the indefinite integral at $a$; thus, $C$ is the initial value if $a$ is the initial point. But for authors who use this concept, there is often no need to mention either $a$ or $C$ (and hence no terminology needed for them), because they are interested only in whether some other function $F$ is an indefinite integral of $f$, where $f$ is a locally integrable function on some closed interval.

Definition

If $F$ is a partial function from $\mathbb{R}$ to $\mathbb{R}$, then $F$ is an antiderivative of $f$ (or an antidifferential of $f \,\mathrm{d}x$) if $f$ is the derivative of $F$ on its domain:

$$\forall\, x \in \dom F,\; f(x) = F'(x) .$$

A posteriori, $F$ must be differentiable.

This is the usual meaning of ‘indefinite integral’ in modern Calculus textbooks using the Riemann integral, especially when the domain of $f$ is an interval.

Definition

If $F$ is a Lebesgue-measurable partial almost function from $\mathbb{R}$ to $\mathbb{R}$, then $F$ is an almost antiderivative of $f$ if $f$ is the derivative of $F$ almost everywhere:

$$\operatorname{ess}\forall\, x \in \dom F,\; f(x) = F'(x) .$$

We are especially interested in the case where $F$ is absolutely continuous.

This is not standard terminology, but it fits in well with other ‘almost’ terminology in measure theory. This is a common meaning of ‘indefinite integral’ when using the Lebesgue integral.

Properties

The main property linking the different kinds of indefinite integral is the fundamental theorem of calculus (FTC). For various definitions of integral, one can prove that every semidefinite integral, or more generally any indefinite integral in the sense of Definition , is an antiderivative; and that every antiderivative, or more generally every almost antiderivative, is an indefinite integral; possibly with technical conditions (depending on the type of integral concerned) such as differentiability or absolute continuity. See that article for details.

Indefinite integrals provide solutions to differential equations. Of course, the definition of an antiderivative is that it is the solution to a particularly simple differential equation. Employing the FTC, we see that the indefinite integrals are the solutions to the corresponding initial-value problems. Specifically, the solution to

$$\frac{\mathrm{d}y}{\mathrm{d}x} = f(x),\; y|_{x=a} = C$$

is the indefinite integral of $f$ with initial point $a$ and initial value $C$:

$$y = C + \int_a f(x) \,\mathrm{d}x .$$

Some more general initial-value problems can be solved similarly; for example, the implicit solution to the separable IVP

$$g(y) \frac{\mathrm{d}y}{\mathrm{d}x} = f(x),\; y|_{x=a} = C$$

is

$$\int_C g(y) \,\mathrm{d}y = \int_a f(x) \,\mathrm{d}x .$$

Similarly, the solution to the linear IVP

$$\frac{\mathrm{d}y}{\mathrm{d}x} + P(x) \,y = Q(x),\; y|_{x=a} = C$$

is

$$y = \frac{C + \int_a Q(x) \,\mu(x) \,\mathrm{d}x}{\mu(x)} ,$$

where $\mu(x) = \exp \int_a P(x) \,\mathrm{d}x$.

Generalizations

In cartesian spaces

When $\omega$ is an exterior $1$-form on a subspace $S$ of the cartesian space $\mathbb{R}^n$, then we can define an antiderivative (or antidifferential) of $\omega$ to be any real-valued function $f$ on $S$ such that $\mathrm{d}f = \omega$. Then when $P$ is a point in $S$ (or perhaps its closure), we can define the value of the semidefinite integral of $\omega$ with initial point $P$ to be the integral of $\omega$ along a straight line segment from $P$; the domain is a star-convex set? radiating from $P$ and contained in (the closure of) $S$. If we define an indefinite integral as a semidefinite integral plus a constant initial value, then every antiderivative of $\omega$ on a star-convex set is an indefinite integral. Conversely, every indefinite integral is an antiderivative if $\omega$ is closed. If $\omega$ is not closed, then it still has indefinite integrals (as long as it's continuous or otherwise locally integrable), but these are no longer antiderivatives (which non-closed forms never have).

This can perhaps be generalized to Riemannian manifolds by considering integrals along geodesics; although the geodesic between two points is not always unique (even when it exists), it is unique on a sufficiently small (and often quite large) neighbourhood. (For example, on a sphere, as long as $\omega$ is integrable, we can define the indefinite integral this way at every point except the one directly opposite the initial point.)

However, this straight-line definition seems rather artificial, and it might be best to use the more notion of semidefinite integral below, applicable only to closed forms but on more general manifolds.

On manifolds

We can generalize to exterior differential forms on differentiable manifolds, generalizing the FTC to the Stokes theorem. It's clear what an antiderivative is in this context: $\alpha$ is an exterior antiderivative of $\omega$ iff $\omega$ is the exterior derivative of $\alpha$. On a smooth manifold, we know what ‘almost’ means and so can also define exterior almost antiderivatives.

If $\omega$ is a $1$-form on any differentiable manifold and $P$ is a point in its domain, then the semidefinite integral of $\omega$ with initial point $P$ is defined at another point $Q$ iff the integral of $\omega$ is the same along any path from $P$ to $Q$ (and then that integral is the value). We can define an indefinite integral by adding a constant initial value. Then every antiderivative on a path-connected domain is an indefinite integral, and conversely every indefinite integral is an antiderivative. By definition, $\omega$ is exact iff an antiderivative exists, and therefore iff an indefinite integral exists on each path-connected component?. Similarly, $\omega$ is closed iff it has an indefinite integral on a neighbourhood of each point; if the domain of $\omega$ is simply connected, then the indefinite integral can be extended to the entire domain.

It remains to consider a good notion of semidefinite and indefinite integrals for higher-rank forms.

In algebraic limit fields

In an algebraic limit field $F$, the notion of a derivative is well defined and is represented by the Newton-Leibniz operator $\tilde{D}$. Given a subset $S \subseteq F$, and a function $f:S \to F$, the set of antiderivatives of $f$ is the fiber of the Newton-Leibniz operator at $f$:

$$\{g \in D^1(S, F) \vert \tilde{D}(g) = f\}$$

An antiderivative is an element of the above subset.

See also

• differentiable function
• integrable function