# Definition:Primitive (Calculus)/Arbitrary Constant

## Definition

Let the real or complex function $F$ be a primitive of the function $f$.

From the language in which it is couched, it is apparent that the primitive of a function may not be unique, otherwise we would be referring to $F$ as ** the primitive** of $f$.

This point is made apparent in Primitives which Differ by Constant: if a function has a primitive, there is an infinite number of them, all differing by a constant.

That is, if $F$ is a primitive for $f$, then so is $F + C$, where $C$ is a constant.

This constant is known as an **arbitrary constant**.

## Also known as

This **arbitrary constant** can also be found being referred to as a **disposable constant** -- that is, it may be "disposed as desired", not "disposed of".

The more neutral term **constant of integration** can also frequently be seen.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 14$: Definition of an Indefinite Integral - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**arbitrary constant**