Antiderivative

A function whose derivative equals a given function.

Antiderivative

An antiderivative (or primitive) of a function $f: I \to \mathbb{R}$ on an interval $I$ is a differentiable function $F: I \to \mathbb{R}$ such that

F'(x) = f(x) \quad \text{for all } x \in I.

Non-uniqueness

If $F$ is an antiderivative of $f$ , then so is $F + C$ for any constant $C$ . Moreover, any two antiderivatives differ by a constant: if $F' = G' = f$ on an interval, then $F - G$ is constant.

The family of all antiderivatives is written as the indefinite integral:

\int f(x)\, dx = F(x) + C.

Connection to definite integrals

The Fundamental Theorem of Calculus states that if $f$ is continuous, then

F(x) = \int_a^x f(t)\, dt

is an antiderivative of $f$ .

Examples

An antiderivative of $x^n$ (for $n \neq -1$ ) is $\frac{x^{n+1}}{n+1}$ .
An antiderivative of $\cos x$ is $\sin x$ .
An antiderivative of $e^x$ is $e^x$ .