Antiderivative¶

Definition¶

Let $I$ be an interval, and let $f\colon I\to\mathbb{R}\text{.}$ A function $F\colon I\to\mathbb{R}$ is an antiderivative of $f$ on $I$ if $F'(x)=f(x)$ at every interior point of $I\text{,}$ with one-sided derivatives at included endpoints.

Interpretation¶

An antiderivative of $f$ is a function $F$ whose derivative is $f$, so finding one reverses differentiation: given the slope at every point, recover a function with those slopes.

The figure shows antiderivatives of $f(x)=x$$x$: the curves $F(x)=\tfrac{x^2}{2}+C$ for $C=-1,0,1,2$ are vertical shifts of one another. At $x=1$ their tangent segments are parallel, all with slope $f(1)=1$$1$. Shifting a graph up or down does not change its slopes, which is why $F+C$ is again an antiderivative.

The derivative is the test for an antiderivative: differentiating $F$ must give $f$. At an included endpoint of $I$ only a one-sided derivative is available, so the condition there is one-sided, while at interior points it is two-sided.

Not every function has an antiderivative. The signum function has none on $[-1,1]$: an antiderivative would be $-x$ plus a constant on $[-1,0)$ and $x$ plus a constant on $(0,1]$, and continuity at $0$ would force it to be $|x|+C$$C$, which is not differentiable at $0$. Every continuous $f$ on $[a,b]$ does have one, the area function $\int_a^x f(t)\,\mathrm{d}t$ of the fundamental theorem of calculus, part 1.

Antiderivatives are what make integrals computable: part 2 of the fundamental theorem evaluates $\int_a^b f(x)\,\mathrm{d}x$ as $F(b)-F(a)$$F(a)$, and the set of all antiderivatives is the indefinite integral.

Key Results¶

• If $F$ is an antiderivative of $f$ on an interval $I\text{,}$ then so is $F + C$ for any constant $C \in \mathbb{R}\text{.}$
• Antiderivatives are not unique; on an interval, any two antiderivatives of $f$ differ by a constant.
• Every continuous function on an interval has an antiderivative, but an integrable function need not have one: the signum function is integrable on $[-1,1]$ and has no antiderivative there.

Examples¶

• If $f(x) = 2x\text{,}$ then $F(x) = x^2$ is an antiderivative of $f\text{.}$
• If $f(x) = \cos x\text{,}$ then $F(x) = \sin x+2$ is an antiderivative of $f\text{.}$

Prerequisites¶

• Endpoint and Bounded Interval
• Derivative and Differentiability

Related Concepts¶

• Riemann Integral
• Fundamental Theorem of Calculus Part 1
• Integration by Parts

Sources¶

• Math I: Calculus · Lecture 11

Aliases: primitive, primitive function Symbols: element of, integral, real numbers

Last updated: 20 October 2026 03:02