Antiderivater

Antiderivater is not a standard term in English; the widely used term is antiderivative (also called a primitive function). An antiderivative of a function f is a function F such that F'(x) = f(x) for all x in the domain. If F is an antiderivative of f, then every function of the form F(x) + C, where C is a constant, is also an antiderivative because (F + C)' = F' = f.

Existence: If f is continuous on an interval, then an antiderivative exists. A common way to define one is F(x) = ∫_a^x f(t) dt for some reference point a. The Fundamental Theorem of Calculus links antiderivatives and derivatives, showing that differentiation and integration are inverse processes.

Examples: The antiderivative of 2x is x^2 + C. The antiderivative of sin x is −cos x + C. Some functions do not have antiderivatives that can be expressed in elementary functions; for example, the antiderivative of e^(−x^2) cannot be written in elementary form, though it can be represented by a definite integral or special functions.

Definite integrals and constant of integration: If F is an antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a). The constant of integration is not fixed by a definite integral, since it cancels in the difference. In cases where no elementary antiderivative exists, numerical methods or special functions are used to evaluate integrals or to describe the antiderivative.

See also: derivative, definite integral, indefinite integral, fundamental theorem of calculus, primitive function.