Antiderivative | Wiskunde op Tilburg University

Definition: A function

$F(x)$ is an antiderivative of a function

$f(x)$ if

$F'(x)=f(x)$ .

Examples:

$F(x)=x^3+2x$ is an antiderivative of $f(x)=3x^2+2$ , because $F'(x)=f(x)$ ;
$F(x)=4e^x$ is an antiderivative of $f(x)=4e^x$ , because $F'(x)=f(x)$ ;
$F(x)=2\sqrt{x+1}$ is an antiderivative of $f(x)=\frac{1}{\sqrt{x+1}}$ , because $F'(x)=f(x)$ .

Note that in Example 1 not only

$F(x)=x^3+2x$ , but also

$F(x)=x^3+2x+4$ ,

$F(x)=x^3+2x-16$ and

$F(x)=x^3+2x - 2\sqrt{5}$ are antiderivatives of

$f(x)=3x^2+2$ . This leads to the following result.

Theorem: If the function

$F(x)$ is an antiderivative of the function

$f(x)$ , then for every constant

$c$ , also

$F(x)+c$ is an antiderivative of the function

$f(x)$ .

Hence, in Example 2 and 3 it holds that

$F(x)=4e^x+c$ and

$F(x)=2\sqrt{x+1}+c$ describe all the antiderivatives of the functions

$f(x)=4e^x$ and

$f(x)=\frac{1}{\sqrt{x+1}}$ , respectively.