How do you find all possible antiderivatives of a function?

Feb 20, 2015

Hello !

If $f$ is function definite on a INTERVAL $I$ (it's very important), and if $F$ is a particular antiderivative of $f$, then all other antiderivative of $f$ are

$F + c$

where $c$ is a real constant.

Proof. If $G$ is another antiderivative, then $G ' = f = F '$, so $\left(G - F\right) ' = f - f = 0$. Because $I$ is an interval, you can say that $G - F$ is constant (so called $c$). Conclusion, $G = F + c$.

Remark. If $I$ is not an interval, like ${\mathbb{R}}^{\setminus} \star$, it's false because you can have $f ' = 0$ with $f$ not constant.

Example : $f \left(x\right) = \setminus \frac{x}{| x |}$ definite on ${\mathbb{R}}^{\setminus} \star$ :
- if $x > 0$, then $f \left(x\right) = 1$
- if $x < 0$, then $f \left(x\right) = -$ - for all x\in RR^\star$,$f'(x) = 0$, b u t$f$i s \neg c o n s \tan t o n$RR^\star#.