Antiderivative

Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$. A function $F \colon I \to \mathbb{R}$ is an antiderivative of $f$ on $I$ if $F$ is differentiable on $I$ and $F'(x) = f(x)$ for all $x \in I$.

Remarks

If $F$ is an antiderivative of $f$, then so is $F + C$ for each constant $C$. On an interval, these are all the antiderivatives, because a function with zero derivative is constant. For example, $x^3/3$ is an antiderivative of $x^2$ on $\mathbb{R}$. A function with a jump has no antiderivative on an interval, by Darboux's theorem.

Explanations

L1 human

An antiderivative undoes differentiation. We look for a function whose derivative is the given function.