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

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

Ordettyreomqu · Accepted Answer

Step 1The statement is equivalent to:       F    1    ′    =      F    2    ′    =  fbut as others have touched on, the equality is not maintained for a definite integral. What this means is that whilst:      ∫                  x        1                            x        2                  F    1      d  x  =      ∫                  x        1                            x        2                  F    2      d  x      F    1    (      x    2    )  −      F    1    (      x    1    )  =      F    2    (      x    2    )  −      F    2    (      x    1    )Step 2this does not imply that either of the following are true:      F    1    (      x    1    )  =      F    2    (      x    1    )        F    1    (      x    2    )  =      F    2    (      x    2    )      F    1    (      x    1    )  =      F    2    (      x    1    )        F    1    (      x    2    )  =      F    2    (      x    2    )which is just the same as how:   2  −  1  =  100  −  99  ⇏  2  =  100This is what it means for functions to differ by a constant, and how two functions can be antiderivatives of the same function

gorgeousgen9487 · Accepted Answer

Step 1It doesn&#039;t mean they cannot be equal. It means they aren&#039;t necessarily equal. If we know that       F    1    ′    (  x  )  =      F    2    ′    (  x  )  =  2  x, then it might be true that      F    1    (  x  )  =      x    2          F    2    (  x  )  =      x    2        F    1    (  x  )  =      x    2          F    2    (  x  )  =      x    2  Step 2But we could also have       F    1    (  x  )  =      x    2    +  π        F    2    (  x  )  =      x    2    −  1000      F    1    (  x  )  =      x    2    +  π        F    2    (  x  )  =      x    2    −  1000So we can&#039;t conclude that       F    1    =      F    2  , but we also cannot prove       F    1    ≠      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&#039;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 &quot;False&quot;. &quot;Is it true that&quot; is often used synonymously with &quot;Does it follow that&quot;, and that&#039;s a shame.

If F 1 </msub> and F 2 </msub> are the antiderivatives of f(

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