Primitives which Differ by a Constant

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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 interval.

Suppose $G$ is a primitive for $f$.

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle D_x \left({ F \left({x}\right) - G \left({x}\right) }\right)$	$=$	$\displaystyle D_x \left({ F \left({x}\right) }\right) - D_x \left({ G \left({x}\right) }\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum Rule for Derivatives
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle f \left({x}\right) - f \left({x}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$F, G$ are a primitives for $f$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle 0$	$\displaystyle $	$\displaystyle $	$\displaystyle $

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

$\Box$

Now suppose $G \left({x}\right) = F \left({x}\right) + c$.

We compute:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle D_x G \left({x}\right)$	$=$	$\displaystyle D_x \left({ F \left({x}\right) + c }\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle D_x \left({ F \left({x}\right) }\right) + 0$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum Rule for Derivatives; Differentiation of a Constant
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle f \left({x}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$F$ is a primitive for $f$

Hence $G$ is also a primitive for $f$.

$\blacksquare$

As there is an uncountable number of possible constants (one for every possible real number), it follows that if a function has a primitive, it has an uncountable number of them.

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle D_x \left({ F \left({x}\right) - G \left({x}\right) }\right)\)	\(=\)	\(\displaystyle D_x \left({ F \left({x}\right) }\right) - D_x \left({ G \left({x}\right) }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum Rule for Derivatives
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle f \left({x}\right) - f \left({x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	$F, G$ are a primitives for $f$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle D_x G \left({x}\right)\)	\(=\)	\(\displaystyle D_x \left({ F \left({x}\right) + c }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle D_x \left({ F \left({x}\right) }\right) + 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum Rule for Derivatives; Differentiation of a Constant
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle f \left({x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	$F$ is a primitive for $f$