Let f be an integrable function on [ a , b ], F be the antiderivative of f. Then ∫ a b f d u = F ( b ) − F ( a )Is it suggesting that while f is integrable, the derivative of the integral may not be f?

Question

Let   f be an integrable function on   [  a  ,  b  ],   F be the antiderivative of   f. Then      ∫    a    b    f    d  u  =  F  (  b  )  −  F  (  a  )Is it suggesting that while   f is integrable, the derivative of the integral may not be   f?

Kaylin Barry · Accepted Answer

Just because a function is integrable, doesn&#039;t mean it has an antiderivative. Consider, for example, the function   f such that   f  (  0  )  =  1 and   f  (  x  )  =  0 for all   x  ≠  0.In this case, the derivative of       ∫    c    x    f  (  t  )  d  t is not equal to   f  (  x  ) at   x  =  0.

Let f be an integrable function on [ a , b ] , F be the antiderivative

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