Primitive functions/Relation to fundamental theorem/Section

Definition

Let ${}I\subseteq \mathbb {R}$ denote an interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function. A function

$F\colon I\longrightarrow \mathbb {R}$

is called a primitive function for ${}f$, if ${}F$ is differentiable on ${}I$ and if ${}F'(x)=f(x)$ holds for all

${}x\in I$.

A primitive function is also called an antiderivative. The fundamental theorem of calculus might be rephrased, in connection with

as an existence theorem for primitive functions.

Corollary

Let ${}I$ denote a real interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a continuous function. Then ${}f$ has a primitive function.

Proof

Let ${}a\in I$ be an arbitrary point. Due to

there exists the function

${}F(x)=\int _{a}^{x}f(t)\,dt\,,$

and because of the Fundamental theorem the identity ${}F'(x)=f(x)$ holds. This means that ${}F$ is a primitive function for ${}f$.

$\Box$

Lemma

Let ${}I$ denote a real interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function. Suppose that ${}F$ and ${}G$ are primitive functions of ${}f$. Then ${}F-G$ is a constant function.

Proof

We have

${}(F-G)'=F'-G'=f-f=0\,.$

Therefore, due to

the difference ${}F-G$ is constant.

$\Box$

Isaac Newton (1643-1727)

Gottfried Wilhelm Leibniz (1646-1716)

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

Corollary

Let ${}I$ denote a real interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a continuous function. Suppose that ${}F$ is a primitive function for ${}f$. Then for ${}a,b\in I$, the identity

${}\int _{a}^{b}f(t)\,dt=F(b)-F(a)\,$

holds.

Proof

Due to

the integral exists. With the integral function

${}G(x):=\int _{a}^{x}f(t)\,dt\,,$

we have the relation

${}\int _{a}^{b}f(t)\,dt=G(b)=G(b)-G(a)\,.$

Because of

the function ${}G$ is differentiable and

${}G'(x)=f(x)\,$

holds. Hence ${}G$ is a primitive function for ${}f$. Due to

we have ${}F(x)=G(x)+c$. Therefore,

${}\int _{a}^{b}f(t)\,dt=G(b)-G(a)=F(b)-c-F(a)+c=F(b)-F(a)\,.$

$\Box$

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

${}\int _{}^{}f(t)\,dt=F+c\,.$

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