# Definition:Primitive (Calculus)/Real

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## Contents

## Definition

Let $F$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

Let $f$ be a real function which is continuous on the open interval $\openint a b$.

Let:

- $\forall x \in \openint a b: \map {F'} x = \map f x$

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

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

- $\displaystyle F = \int \map f x \rd x$

## 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: Calculus