Primitives which Differ by Constant

Theorem
Let $F$ be a primitive for a real function $f$ on the closed interval $\left[{a \,. \, . \, b}\right]$.

Let $G$ be a real function defined on $\left[{a \,. \, . \, b}\right]$.

Then $G$ is a primitive for $f$ on $\left[{a \,. \, . \, b}\right]$ iff:
 * $\exists c \in \R: \forall x \in \left[{a \, . \, . \, b}\right]: G \left({x}\right) = F \left({x}\right) + c$

That is, iff $F$ and $G$ differ by a constant on the whole length of the interval.

Proof
From Differentiation of a Constant and Sum Rule for Derivatives it is clear that $F \left({x}\right) + c$ is a primitive for $f$.

So, suppose $G$ is a primitive for $f$.

Then $F - G$ is continuous on $\left[{a \,. \, . \, b}\right]$, differentiable on $\left({a \, . \, . \, b}\right)$, and:


 * $\forall x \in \left[{a \, . \, . \, b}\right]: D \left({F \left({x}\right) - G \left({x}\right)}\right) = F^{\prime} \left({x}\right) - G^{\prime} \left({x}\right) = f \left({x}\right) - f \left({x}\right) = 0$.

From Zero Derivative means Constant Function it follows that $F - G$ is constant on $\left[{a \,. \, . \, b}\right]$, hence the result.