Primitivifunktion

A primitivifunktion, also known as an antiderivative or indefinite integral, is a fundamental concept in calculus. It is a function whose derivative is equal to a given function. In other words, if F(x) is a primitivifunktion of f(x), then the derivative of F(x) with respect to x is f(x), denoted as F'(x) = f(x).

Primitivifunktionen are not unique; if F(x) is a primitivifunktion of f(x), then so is F(x) + C, where C is any constant. This is because the derivative of a constant is zero. Therefore, primitivifunktionen are often expressed with the constant of integration, denoted as F(x) + C.

The process of finding a primitivifunktion is called integration. There are several methods for finding primitivifunktionen, including substitution, integration by parts, and using tables of integrals. Some functions, however, do not have a primitivifunktion that can be expressed in terms of elementary functions.

Primitivifunktionen have numerous applications in mathematics and physics, including calculating areas under curves, volumes of solids of revolution, and solving differential equations. They are also used in the study of complex analysis, where they are known as holomorphic functions.