Definition:Primitive (Calculus)

Definition
Let $$F$$ be a real function which is continuous on the closed interval $$\left[{a \,. \, . \, b}\right]$$ and differentiable on the open interval $$\left({a \, . \, . \, b}\right)$$.

Let $$f$$ be a real function which is continuous on the open interval $$\left({a \, . \, . \, b}\right)$$.

Suppose that:
 * $$\forall x \in \left({a \, . \, . \, b}\right): F^{\prime} \left({x}\right) = f \left({x}\right)$$.

Then $$F$$ is known as a primitive (or an antiderivative) of $$f$$.

Integration
The process of finding a primitive for a function is known as integration.

Arbitrary Constant
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 a 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 (or disposable constant - that is, it may be "disposed as desired", not "disposed of").

Indefinite Integral
From the Fundamental Theorem of Calculus, it is apparent that to find the value of a definite integral for a function between two points, one can find the value of the primitive of the function at those points and subtract one from the other.

Thus arises the notation $$\int f \left({x}\right) dx = F \left({x}\right) + C$$.

In this context, the expression $$\int f \left({x}\right) dx$$ is known as the indefinite integral of $$f$$.