Primitives which Differ by Constant

Theorem

Let $F$ be a primitive for a real function $f$ on the closed interval $\closedint a b$.

Let $G$ be a real function defined on $\closedint a b$.

Then $G$ is a primitive for $f$ on $\closedint a b$ if and only if:

$\exists c \in \R: \forall x \in \closedint a b: \map G x = \map F x + c$

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

Let $f$ be an integrable function on the closed interval $\closedint a b$.

Then there exist an uncountable number of primitives for $f$ on $\closedint a b$.

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

Then $F - G$ is continuous on $\closedint a b$, differentiable on $\openint a b$, and for any $x \in \openint a b$, we have:

$\ds \map {D_x} {\map F x - \map G x}$

$=$

$\ds \map {D_x} {\map F x} - \map {D_x} {\map G x}$

$\ds $

$=$

$\ds \map f x - \map f x$

$F, G$ are a primitives for $f$

$\ds $

$=$

$\ds 0$

From Zero Derivative implies Constant Function it follows that $F - G$ is constant on $\closedint a b$.

Hence the result.

$\Box$

Now suppose $\map G x = \map F x + c$.

We compute:

$\ds D_x \map G x$

$=$

$\ds \map {D_x} {\map F x + c}$

$\ds $

$=$

$\ds \map {D_x} {\map F x} + 0$

$\ds $

$=$

$\ds \map f x$

$F$ is a primitive for $f$

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

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Indefinite Integrals: Definition of an Indefinite Integral
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.11$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integration
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integration
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: Definition of an Indefinite Integral
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): antiderivative
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): antiderivative