# Antiderivative

Definition: A function $F(x)$ is an antiderivative of a function $f(x)$ if $F'(x)=f(x)$.

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