# Primitive functions/Relation to fundamental theorem/Section

Let denote an interval, and let

denote a function. A function

is called a *primitive function* for , if is
differentiable
on and if
holds for all

A primitive function is also called an *antiderivative*. The fundamental theorem of calculus might be rephrased, in connection with
fact,
as an existence theorem for primitive functions.

Let be an arbitrary point. Due to fact, there exists the function

and because of the Fundamental theorem, the identity holds. This means that is a primitive function for .

Let denote a real interval, and let

denote a function. Suppose that and are primitive functions of . Then is a constant function.

The following statement is also a version of the fundamental theorem, it is called the *Newton-Leibniz-formula*.

Let denote a real interval, and let

denote a continuous function. Suppose that is a primitive function for . Then for , the identity

holds.

Due to fact, the integral exists. With the integral function

we have the relation

Because of fact, the function is differentiable and

holds. Hence is a primitive function for . Due to fact, we have . Therefore,

Since a primitive function is only determined up to an additive constant, we sometimes write

Here is called a *constant of integration*. In certain situations, in particular in relation with *differential equations*, this constant is determined by further conditions.