# Definition:Primitive (Calculus)/Real

## Contents

## Definition

Let $F$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$ and differentiable on the open interval $\left({a \,.\,.\, b}\right)$.

Let $f$ be a real function which is continuous on the open interval $\left({a \,.\,.\, b}\right)$.

Let:

- $\forall x \in \left({a \,.\,.\, b}\right): F' \left({x}\right) = f \left({x}\right)$

where $F'$ denotes the derivative of $F$ with respect to $x$.

Then $F$ is **a primitive of $f$**, and is denoted:

- $\displaystyle F = \int f \left({x}\right) \, \mathrm d x$

## Also known as

A **primitive** is also known as an **antiderivative**.

The term **indefinite integral** is also popular.

## Also see

- Results about
**integral calculus**can be found here.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 14$: Definition of an Indefinite Integral - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 13.9$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World