Definition:Primitive (Calculus)/Constant of Integration

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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 the constant of integration.


Examples

Primitive of $2 x$

Consider the primitive:

$\ds \int 2 x \rd x = x^2 + c$

the symbol $c$ represents the (arbitrary) constant of integration.

Its value can be determined by a boundary condition.


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.


Sources

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