Nylah Hendrix

2022-07-04

If ${F}_{1}$ and ${F}_{2}$ are the antiderivatives of f(x), then ${F}_{1}={F}_{2}$?

Ordettyreomqu

Expert

Step 1
The statement is equivalent to: ${F}_{1}^{\prime }={F}_{2}^{\prime }=f$
but as others have touched on, the equality is not maintained for a definite integral. What this means is that whilst:
${\int }_{{x}_{1}}^{{x}_{2}}{F}_{1}\phantom{\rule{thinmathspace}{0ex}}dx={\int }_{{x}_{1}}^{{x}_{2}}{F}_{2}\phantom{\rule{thinmathspace}{0ex}}dx$
${F}_{1}\left({x}_{2}\right)-{F}_{1}\left({x}_{1}\right)={F}_{2}\left({x}_{2}\right)-{F}_{2}\left({x}_{1}\right)$
Step 2
this does not imply that either of the following are true:
${F}_{1}\left({x}_{1}\right)={F}_{2}\left({x}_{1}\right)\phantom{\rule{0ex}{0ex}}{F}_{1}\left({x}_{2}\right)={F}_{2}\left({x}_{2}\right)$
${F}_{1}\left({x}_{1}\right)={F}_{2}\left({x}_{1}\right)\phantom{\rule{0ex}{0ex}}{F}_{1}\left({x}_{2}\right)={F}_{2}\left({x}_{2}\right)$
which is just the same as how: $2-1=100-99⇏2=100$
This is what it means for functions to differ by a constant, and how two functions can be antiderivatives of the same function

gorgeousgen9487

Expert

Step 1
It doesn't mean they cannot be equal. It means they aren't necessarily equal. If we know that ${F}_{1}^{\prime }\left(x\right)={F}_{2}^{\prime }\left(x\right)=2x$, then it might be true that
${F}_{1}\left(x\right)={x}^{2}\phantom{\rule{0ex}{0ex}}{F}_{2}\left(x\right)={x}^{2}$
${F}_{1}\left(x\right)={x}^{2}\phantom{\rule{0ex}{0ex}}{F}_{2}\left(x\right)={x}^{2}$
Step 2
But we could also have ${F}_{1}\left(x\right)={x}^{2}+\pi \phantom{\rule{0ex}{0ex}}{F}_{2}\left(x\right)={x}^{2}-1000$
${F}_{1}\left(x\right)={x}^{2}+\pi \phantom{\rule{0ex}{0ex}}{F}_{2}\left(x\right)={x}^{2}-1000$
So we can't conclude that ${F}_{1}={F}_{2}$, but we also cannot prove ${F}_{1}\ne {F}_{2}$.
If this was a true / false question, then technically the statement ${F}_{1}={F}_{2}$ is undecided. But by my powers of reading the minds of problem authors, they didn't mean to ask whether ${F}_{1}$ is equal to ${F}_{2}$, but whether ${F}_{1}$ is necessarily equal to ${F}_{2}$ (this is just an implicit part to many of these problems that really, really ought to be explicit), which means that the intended answer is "False". "Is it true that" is often used synonymously with "Does it follow that", and that's a shame.

Do you have a similar question?