Definition:Primitive (Calculus)/Constant of Integration

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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 the constant of integration.

Also known as

The term constant of integration is often referred to inprecisely as an arbitrary constant.

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

Also see

  • Results about constants of integration can be found here.