In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function. This can be stated symbolically as. The process of solving for antiderivatives is called antidifferentiation and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. The discrete equivalent of the notion of antiderivative is antidifference.
Examples
The function is an antiderivative of, as the derivative of is. As the derivative of a constant is zero, will have an infinite number of antiderivatives, such as, etc. Thus, all the antiderivatives of can be obtained by changing the value of in, where is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other; each graph's vertical location depending upon the value. More generally, the power function has antiderivative if, and if. In physics, the integration of acceleration yields velocity plus a constant. The constant is the initial velocity term that would be lost upon taking the derivative of velocity because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion.
Uses and properties
Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if is an antiderivative of the integrable function over the interval, then: Because of this, each of the infinitely many antiderivatives of a given function is sometimes called the "general integral" or "indefinite integral" of f and is written using the integral symbol with no bounds: If is an antiderivative of, and the function is defined on some interval, then every other antiderivative of differs from by a constant: there exists a number such that for all. is called the constant of integration. If the domain of is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals. For instance is the most general antiderivative of on its natural domain Every continuous function has an antiderivative, and one antiderivative is given by the definite integral of with variable upper boundary: Varying the lower boundary produces other antiderivatives. This is another formulation of the fundamental theorem of calculus. There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions. Examples of these are From left to right, the first four are the error function, the Fresnel function, the trigonometric integral, and the logarithmic integral function. See alsoDifferential Galois theory for a more detailed discussion.
Techniques of integration
Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives. For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions. See the articles on elementary functions and nonelementary integral for further information. There are various methods available:
Inverse function integration, a formula that expresses the antiderivative of the inverse of an invertible and continuous function in terms of the antiderivative of and of.
when integrating multiple times, certain additional techniques can be used, see for instance double integrals and polar coordinates, the Jacobian and the Stokes' theorem
if a function has no elementary antiderivative, its definite integral can be approximated using numerical integration
it is often convenient to algebraically manipulate the integrand such that other integration techniques, such as integration by substitution, may be used.
to calculate the repeated antiderivative of a function, Cauchy's formula is useful :
Computer algebra systems can be used to automate some or all of the work involved in the symbolic techniques above, which is particularly useful when the algebraic manipulations involved are very complex or lengthy. Integrals which have already been derived can be looked up in a table of integrals.
Of non-continuous functions
Non-continuous functions can have antiderivatives. While there are still open questions in this area, it is known that:
Some highly pathological functions with large sets of discontinuities may nevertheless have antiderivatives.
In some cases, the antiderivatives of such pathological functions may be found by Riemann integration, while in other cases these functions are not Riemann integrable.
Assuming that the domains of the functions are open intervals:
A necessary, but not sufficient, condition for a function to have an antiderivative is that have the intermediate value property. That is, if is a subinterval of the domain of and is any real number between and, then there exists a between and such that. This is a consequence of Darboux's theorem.
The set of discontinuities of must be a meagre set. This set must also be an F-sigma set. Moreover, for any meagre F-sigma set, one can construct some function having an antiderivative, which has the given set as its set of discontinuities.
If has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration in the sense of Lebesgue. In fact, using more powerful integrals like the Henstock–Kurzweil integral, every function for which an antiderivative exists is integrable, and its general integral coincides with its antiderivative.
If has an antiderivative on a closed interval, then for any choice of partition if one chooses sample points as specified by the mean value theorem, then the corresponding Riemann sumtelescopes to the value.
