Definition:Primitive (Calculus)
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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:
- $\forall x \in \left({a .. b}\right): F^{\prime} \left({x}\right) = f \left({x}\right)$
where $F^{\prime}$ signifies the derivative of $F$.
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:
- $\displaystyle \int f \left({x}\right) \ \mathrm d x = F \left({x}\right) + C$
In this context, the expression $\displaystyle \int f \left({x}\right) \ \mathrm d x$ is known as the indefinite integral of $f$.
