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).