calculus, an antiderivative, primitive or indefinite integral [
Antiderivatives are also called general integrals, and sometimes integrals. The latter term is generic, and refers not only to indefinite integrals (antiderivatives), but also to definite integrals. When the word integral is used without additional specification, the reader is supposed to deduce from the context whether it is referred to a definite or indefinite integral. Some authors define the indefinite integral of a function as the set of its infinitely many possible antiderivatives. Others define it as an arbitrarily selected element of that set. Wikipedia adopts the latter approach.] of a function "f" is a function "F" whose
derivativeis equal to "f", i.e., "F" ′ = "f". The process of solving for antiderivatives is antidifferentiation (or indefinite integration). Antiderivatives are related to definite integrals through the fundamental theorem of calculus, and provide a convenient means for calculating the definite integrals of many functions.
The function "F"("x") = "x"3/3 is an antiderivative of "f"("x") = "x"2. As the derivative of a
constantis zero, "x"2 will have an infinite number of antiderivatives; such as ("x"3/3) + 0, ("x"3 / 3) + 7, ("x"3 / 3) − 42, etc. Thus, the entire antiderivative family of "x"2 can be obtained by changing the value of C in "F"("x") = ("x"3 / 3) + "C"; where "C" 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 location depending upon the value of "C".
Uses and properties
Antiderivatives are important because they can be used to compute definite integrals, using the
fundamental theorem of calculus: if "F" is an antiderivative of the integrable function "f", then:
Because of this, each of the infinitely many antiderivatives of a given function "f" is sometimes called the "general integral" or "indefinite integral" of "f" and is written using the integral symbol with no bounds::
If "F" is an antiderivative of "f", and the function "f" is defined on some interval, then every other antiderivative "G" of "f" differs from "F" by a constant: there exists a number "C" such that "G"("x") = "F"("x") + "C" for all "x". "C" is called the
arbitrary constant of integration. If the domain of "F" is a disjoint unionof 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
continuous function"f" has an antiderivative, and one antiderivative "F" is given by the definite integral of "f" with variable upper boundary::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 functions whose antiderivatives, even though they exist, cannot be expressed in terms of
elementary functions (like polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functionsand their combinations). Examples of these are :
differential Galois theoryfor 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 article on elementary functions for further information.
We have various methods at our disposal:
linearity of integrationallows us to break complicated integrals into simpler ones
integration by substitution, often combined with trigonometric identities or the natural logarithm
integration by partsto integrate products of functions
inverse chain rule method, a special case of integration by substitution
* the method of
partial fractions in integrationallows us to integrate all rational functions (fractions of two polynomials)
* integrals can also be looked up in a
table of integrals
* when integrating multiple times, we can use certain additional techniques, see for instance
double integrals and polar coordinates, the Jacobianand the Stokes' theorem
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
* if a function has no elementary antiderivative (for instance, exp("x"2)), its definite integral can be approximated using
* to calculate the ( times) repeated antiderivative of a function Cauchy's formula is useful: :
Antiderivatives of non-continuous functions
To illustrate some of the subtleties of the fundamental theorem of calculus, it is instructive to consider what kinds of non-continuous functions might 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.
We first state some general facts and then provide some illustrative examples. Throughout, we assume that the domains of our functions are open intervals.
* A necessary, but not sufficient, condition for a function "f" to have an antiderivative is that "f" have the intermediate value property. That is, if ["a","b"] is a subinterval of the domain of "f" and "d" is any real number between "f"("a") and "f"("b"), then "f"("c")="d" for some "c" between "a" and "b". To see this, let "F" be an antiderivative of "f" and consider the continuous function "g"("x")="F"("x")-"dx" on the closed interval ["a", "b"] . Then "g" must have either a maximum or minimum "c" in the open interval ("a","b") and so 0="g"′("c")="f"("c")-"d".
* The set of discontinuities of "f" must be a meagre set. This set must also be an
F-sigmaset (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 "f" having an antiderivative, which has the given set as its set of discontinuities.
* If "f" has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of
Lebesgue measure0, then an antiderivative may be found by integration.
* If "f" has an antiderivative "F" on a closed interval ["a","b"] , then for any choice of partition
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