Antiderivative


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 often denoted by capital Roman letters such as and.
Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
In physics, antiderivatives arise in the context of rectilinear motion. The discrete equivalent of the notion of antiderivative is antidifference.

Examples

The function is an antiderivative of, since the derivative of is. Since 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. The graphs of antiderivatives of a given function are vertical translations of each other, with 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. Thus, integration produces the relations of acceleration, velocity and displacement:

Uses and properties

Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if is an antiderivative of the continuous function over the interval, then:
Because of this, each of the infinitely many antiderivatives of a given function may be called the "indefinite integral" of f and 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:
for any in the domain of. Varying the lower boundary produces other antiderivatives, but not necessarily all possible antiderivatives. This is another formulation of the fundamental theorem of calculus.
There are many elementary functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions. Elementary functions are polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations under composition and linear combination. Examples of these nonelementary integrals are
For a more detailed discussion, see also Differential Galois theory.

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. To learn more, see elementary functions and nonelementary integral.
There exist many properties and techniques for finding antiderivatives. These include, among others:
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 sum telescopes to the value. However, if is unbounded, or if is bounded but the set of discontinuities of has positive Lebesgue measure, a different choice of sample points may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.

    Some examples

Basic formulae

  • If, then.
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