Is there a formal definition for antiderivatives?In the way the derivative can be defined as a limit, specifically f ′ ( x ) := lim h → 0 f ( x + h ) − f ( x ) h or any of the other possible variants, is there a way to define the antiderivative, as in indefinite integral?The handful of sources I've looked over (Wikipedia and MathWorld, to name a few) all refer to the antiderivative simply as a "nonunique inverse operator" (I'm paraphrasing). I can't say I'm completely satisfied with this notion. Is the following the best we can do?If F(x) is a function that satisfies d d x F ( x ) = f ( x ), then F(x) is called an antiderivative of f(x). (Forgive the lack of formality.)

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Is there a formal definition for antiderivatives?In the way the derivative can be defined as a limit, specifically      f    ′    (  x  )  :=      lim          h      →      0                  f      (      x      +      h      )      −      f      (      x      )        h   or any of the other possible variants, is there a way to define the antiderivative, as in indefinite integral?The handful of sources I&#039;ve looked over (Wikipedia and MathWorld, to name a few) all refer to the antiderivative simply as a &quot;nonunique inverse operator&quot; (I&#039;m paraphrasing). I can&#039;t say I&#039;m completely satisfied with this notion. Is the following the best we can do?If F(x) is a function that satisfies             d              d        x              F  (  x  )  =  f  (  x  ), then F(x) is called an antiderivative of f(x). (Forgive the lack of formality.)

Mathias Patrick · Accepted Answer

Step 1Since the antiderivative is so far from unique, we can&#039;t give anything like a pointwise formula for it as it looks like you want without choosing one. Probably the best idea is the definite integral,  F  (  x  )  =      ∫    0    x    f  (  x  )  d  x  =      lim          n      →      ∞            x    n        (          ∑              k        =        0                    n        −        1              f    (    k    x          /        n    )    )  Step 2Here I&#039;ve used the left Riemann sum, which is of course not the only way to define an integral, but it works whenever the integral exists. It&#039;s possible to write a less arbitrary definition, but uglier.

Is there a formal definition for antiderivatives? In the way the derivative can be defined as a lim

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