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Wikipedia / 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 (or indefinite integration), 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 integralss 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 (e.g., in explaining the relationship between position, velocity and acceleration). The discrete equivalent of the notion of antiderivative is antidifference.

Examples

. Since the derivative of a constant is zero will have an infinite can be obtained by changing the value of , 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 if 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 (position, velocity, acceleration, and so on). 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

Because of this, each of the infinitely many antiderivatives of a given function may be called the "indefinite integral" of f

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 for all . is called the constant of integration. If the domain of is a disjoint union

Every continuous function has an antiderivative, and one antiderivative is given by the definite integral of 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 (indeed, there is no pre-defined method for computing indefinite integrals). 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 (since the set of discontinuities of any function must be of this type). 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 as specified by the mean value theorem, then the corresponding Riemann sum telescopes However, if is unbounded, or if is bounded but the set of discontinuities of may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.

Some examples

Basic formulae

  • .

See also

Notes

References

Further reading

  • Introduction to Classical Real Analysis, by Karl R. Stromberg; Wadsworth, 1981 (see also)

  • by Dave L. Renfro