nLab
indefinite integral

Indefinite integrals

Indefinite integrals

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 two 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 ff be a partial function from \mathbb{R} to \mathbb{R}; typically, the domain of ff will be an interval, but we do not require this.

Definition

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

x a xf(t)dt. x \mapsto \int_a^x f(t) \,\mathrm{d}t .

(If x<ax \lt a, then we must define a x\int_a^x as x a-\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 aa and contained in the domain of ff (or at least in its closure if we allow improper integral?s or integrating almost functions). If we start by defining ff as a locally integrable function? on a closed interval II containing aa, then the semidefinite integral will also have II as its domain.

The value at xx of the semidefinite integral from aa may be denoted

af(x)dx \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 tt to unfold the definition. (Indeed, some writers will abuse notation, writing a xf(x)dx\int_a^x f(x) \,\mathrm{d}x for the semidefinite integral.) But as in dy/dx\mathrm{d}y/\mathrm{d}x, the xx 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 af(x)dx\int_a f(x) \,\mathrm{d}x when xx is some number bb, then the notation for this is a bf(x)dx\int_a^b f(x) \,\mathrm{d}x, in which xx has now become a dummy variable but has not simply been replaced with bb.

Definition

If aa and CC are real numbers (with aa in the domain of ff or its closure), then the indefinite integral of ff from aa with initial value CC is the function

xC+ a xf(t)dt. x \mapsto C + \int_a^x f(t) \,\mathrm{d}t .

We may write this value as C+ af(x)dxC + \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 CC is the value of the indefinite integral at aa; thus, CC is the initial value if aa is the initial point. But for authors who use this concept, there is often no need to mention either aa or CC (and hence no terminology needed for them), because they are interested only in whether some other function FF is an indefinite integral of ff, where ff is a locally integrable function on some closed interval.

Definition

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

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

A posteriori, FF must be differentiable.

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

Definition

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

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

We are especially interested in the case where FF 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

F(x)=f(x),F(a)=C F'(x) = f(x),\; F(a) = C

is the indefinite integral of ff with initial point aa and initial value CC:

F(x)=C+ af(x)dx. F(x) = C + \int_a f(x) \,\mathrm{d}x .

On manifolds

If we think of the real line as the prototypical 11-dimensional differentiable manifold and f(x)dxf(x) \,\mathrm{d}x as a differential form on that manifold, then we can try to generalize this to other exterior differential forms, generalizing the FTC to the Stokes theorem. It's clear what an antiderivative is in this context: α\alpha is an exterior antiderivative of ω\omega iff ω\omega is the exterior derivative of α\alpha. On a smooth manifold, we know what ‘almost’ means and so can also define exterior almost antiderivatives. However, it's less clear what a semidefinite integral or indefinite integral should be.

When ω\omega is an exterior 11-form on a subspace of n\mathbb{R}^n and PP is a point in its domain, then we can define the value of the semidefinite integral of ω\omega with initial point PP to be the integral of ω\omega along a straight line segment from PP; the domain is a star-convex set? radiating from PP and contained in domω\dom \omega. If we define an indefinite integral as a semidefinite integral plus a constant initial value, then every antiderivative of ω\omega on a star-convex set is an indefinite integral. Conversely, every indefinite integral is an antiderivative if ω\omega is closed. 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 ω\omega is integrable, we can define the indefinite integral this way at every point except the one directly opposite the initial point.)

On a more general manifold, we need a definition of semidefinite integral that exists much less often, but the value is that we no longer have to assume that ω\omega is closed for half of the FTC; so this is probably the better definition. So, if ω\omega is a 11-form on any differentiable manifold and PP is a point in its domain, then the semidefinite integral of ω\omega with initial point PP is defined at another point QQ iff the integral of ω\omega is the same along any path from PP to QQ (and then that integral is the value). Then we can define an indefinite integral by adding a constant initial value. It's still true that every antiderivative is an indefinite integral (at least on a path-connected domain, or more generally when the domain is a manifold in which we can choose an initial point in each connected component), and now it's also true that every indefinite integral is an antiderivative. By definition, ω\omega is exact iff an antiderivative exists, and therefore iff an indefinite integral exists. Similarly, ω\omega is closed iff it has an indefinite integral on a neighbourhood of each point; if the domain of ω\omega is simply connected, then the indefinite integral can be extended to the entire domain.

Last revised on April 15, 2021 at 17:07:41. See the history of this page for a list of all contributions to it.