Antiderivative

In calculus, an antiderivative of a given real valued function f is a function F whose derivative is equal to f, i.e. F ' = f. For example, F(x) = 1/3 x³ is an antiderivative of f(x) = x².

Antiderivatives are important because they can be used to compute integrals, using the fundamental theorem of calculus: if F is an antiderivative of f, then ∫_a^b f(x) dx = F(b) - F(a).

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.

Every continuous function f has an antiderivative, and one antiderivative is given by the integral of f, another formulation of the fundamental theorem of calculus. There are also some non-continuous functions which have an antiderivative, for example f(x) = 2x sin (1/x) - cos(1/x) with f(0) = 0 is not continuous at x = 0 but has the antiderivative F(x) = x² sin(1/x) with F(0) = 0.

Finding antiderivatives is considerably harder than finding derivatives. One usually consults a table of integrals and constructs antiderivatives for more complicated functions with the techniques of [integration by parts]? and [integration by substitution]?. However, there are many easily described functions whose antiderivatives cannot be expressed in terms of elementary functions at all; an example is f(x) = exp(x²) (see exponential function).