Antiderivative

For lists of antiderivatives of primitive functions, see listsof integrals.In calculus, an antiderivative, primitive function,

The slope eld of F(x) = (x3/3)-(x2/2)-x+c, showing three ofthe innitely many solutions that can be produced by varying thearbitrary constant C.

primitive integral or indenite integral[1] of a functionf is a dierentiable function F whose derivative is equalto the original function f. This can be stated symbolicallyas F = f.[2][3] The process of solving for antiderivativesis called antidierentiation (or indenite integration)and its opposite operation is called dierentiation, whichis the process of nding a derivative.Antiderivatives are related to denite integrals throughthe fundamental theorem of calculus: the denite inte-gral of a function over an interval is equal to the dier-ence between the values of an antiderivative evaluated atthe endpoints of the interval.The discrete equivalent of the notion of antiderivative isantidierence.

1 ExampleThe function F(x) = x3/3 is an antiderivative of f(x) =x2. As the derivative of a constant is zero, x2 will havean innite number of antiderivatives, such as x3/3, x3/3 +1, x3/3 - 2, etc. Thus, all the antiderivatives of x2 can beobtained by changing the value of C in F(x) = x3/3 + C;

where C is an arbitrary constant known as the constant ofintegration. Essentially, the graphs of antiderivatives of agiven function are vertical translations of each other; eachgraphs vertical location depending upon the value of C.In physics, the integration of acceleration yields velocityplus a constant. The constant is the initial velocity termthat would be lost upon taking the derivative of velocitybecause the derivative of a constant term is zero. Thissame pattern applies to further integrations and deriva-tives of motion (position, velocity, acceleration, and soon).

2 Uses and propertiesAntiderivatives are important because they can be used tocompute denite integrals, using the fundamental theo-rem of calculus: if F is an antiderivative of the integrablefunction f and f is continuous over the interval [a, b],then:

Z ba

f(x) dx = F (b) F (a):

Because of this, each of the innitely many antideriva-tives of a given function f is sometimes called the gen-eral integral or indenite integral of f and is writtenusing the integral symbol with no bounds:

Zf(x) dx:

If F is an antiderivative of f, and the function f is denedon some interval, then every other antiderivative G of fdiers from F by a constant: there exists a number C suchthat G(x) = F(x) + C for all x. C is called the arbitraryconstant of integration. If the domain of F is a disjointunion of two or more intervals, then a dierent constantof integration may be chosen for each of the intervals.For instance

F (x) =

( 1x + C1 x < 0 1x + C2 x > 0

is the most general antiderivative of f(x) = 1/x2 on itsnatural domain (1; 0) [ (0;1):

1

2 4 ANTIDERIVATIVES OF NON-CONTINUOUS FUNCTIONS

Every continuous function f has an antiderivative, andone antiderivative F is given by the denite integral off with variable upper boundary:

F (x) =

Z x0

f(t) dt:

Varying the lower boundary produces other antideriva-tives (but not necessarily all possible antiderivatives).This is another formulation of the fundamental theoremof calculus.There are many functions whose antiderivatives, eventhough they exist, cannot be expressed in terms ofelementary functions (like polynomials, exponentialfunctions, logarithms, trigonometric functions, inversetrigonometric functions and their combinations). Exam-ples of these are

Zex

2 dx;Z

sinx2 dx;Z sinx

xdx;

Z1

lnx dx;Z

xx dx:

From left to right, the rst four are the error function,the Fresnel function, the trigonometric integral, and thelogarithmic integral function.See also Dierential Galois theory for a more detaileddiscussion.

3 Techniques of integrationFinding antiderivatives of elementary functions is oftenconsiderably harder than nding their derivatives. Forsome elementary functions, it is impossible to nd an an-tiderivative in terms of other elementary functions. Seethe article on elementary functions for further informa-tion.There are various methods available:

the linearity of integration allows us to break com-plicated integrals into simpler ones

integration by substitution, often combined withtrigonometric identities or the natural logarithm the inverse chain rule method, a special caseof integration by substitution

integration by parts to integrate products of func-tions

Inverse function integration, a formula that ex-presses the antiderivative of the inverse f1 of aninvertible and continuous function f in terms of theantiderivative of f and of f1 .

the method of partial fractions in integration allowsus to integrate all rational functions (fractions of twopolynomials)

the Risch algorithm when integrating multiple times, certain additionaltechniques can be used, see for instance double in-tegrals and polar coordinates, the Jacobian and theStokes theorem

if a function has no elementary antiderivative (forinstance, exp(-x2)), its denite integral can be ap-proximated using numerical integration

it is often convenient to algebraically manipulatethe integrand such that other integration techniques,such as integration by substitution, may be used.

to calculate the (n times) repeated antiderivative ofa function f, Cauchy's formula is useful (cf. Cauchyformula for repeated integration):

Z xx0

Z x1x0

: : :

Z xn1x0

f(xn) dxn : : : dx2 dx1 =Z xx0

f(t)(x t)n1(n 1)! dt:

Computer algebra systems can be used to automate someor all of the work involved in the symbolic techniquesabove, which is particularly useful when the algebraic ma-nipulations involved are very complex or lengthy. Inte-grals which have already been derived can be looked upin a table of integrals.

4 Antiderivatives of non-continuous functions

Non-continuous functions can have antiderivatives.While there are still open questions in this area, it isknown that:

Some highly pathological functions with large setsof discontinuities may nevertheless have antideriva-tives.

In some cases, the antiderivatives of such pathologi-cal functions may be found by Riemann integration,while in other cases these functions are not Riemannintegrable.

Assuming that the domains of the functions are open in-tervals:

A necessary, but not sucient, condition for a func-tion f to have an antiderivative is that f have theintermediate value property. That is, if [a, b] is asubinterval of the domain of f andC is any real num-ber between f(a) and f(b), then f(c) = C for some cbetween a and b. To see this, let F be an antideriva-tive of f and consider the continuous function

4.1 Some examples 3

g(x) = F (x) Cxon the closed interval [a, b]. Then g must have either amaximum or minimum c in the open interval (a, b) andso

0 = g0(c) = f(c) C:

The set of discontinuities of f must be a meagre set.This set must also be an F-sigma set (since the setof discontinuities of any function must be of thistype). Moreover for any meagre F-sigma set, onecan construct some function f having an antideriva-tive, which has the given set as its set of discontinu-ities.

If f has an antiderivative, is bounded on closed nitesubintervals of the domain and has a set of discon-tinuities of Lebesgue measure 0, then an antideriva-tive may be found by integration in the sense ofLebesgue. In fact, using more powerful integralslike the HenstockKurzweil integral, every functionfor which an antiderivative exists is integrable, andits general integral coincides with its antiderivative.

If f has an antiderivative F on a closed interval[a,b], then for any choice of partition a = x0