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