# 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 $\left[{a \,.\,.\, b}\right]$ if and only if:

- $\exists c \in \R: \forall x \in \left[{a \,.\,.\, b}\right]: G \left({x}\right) = F \left({x}\right) + c$

That is, if and only if $F$ and $G$ differ by a constant on the whole interval.

## Proof

### Necessary Condition

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:

\(\displaystyle D_x \left({ F \left({x}\right) - G \left({x}\right) }\right)\) | \(=\) | \(\displaystyle D_x \left({ F \left({x}\right) }\right) - D_x \left({ G \left({x}\right) }\right)\) | Sum Rule for Derivatives | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle f \left({x}\right) - f \left({x}\right)\) | $F, G$ are a primitives for $f$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0\) |

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

$\Box$

### Sufficient Condition

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

We compute:

\(\displaystyle D_x G \left({x}\right)\) | \(=\) | \(\displaystyle D_x \left({ F \left({x}\right) + c }\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle D_x \left({ F \left({x}\right) }\right) + 0\) | Sum Rule for Derivatives and Derivative of Constant | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle f \left({x}\right)\) | $F$ is a primitive for $f$ |

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

$\blacksquare$

## Notes

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.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 14$: Definition of an Indefinite Integral - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 13.11$