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Very often on this forum I have seen the statement "integration and differentiation are inverse of each other".

Is that what Fundamental theorem of calculus is about? Does non-Riemann integrable or everywhere continuous nowhere differentiable functions require modification to FTC or Differentiation is inverse of integration statements?

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The word "integration" is used in two different meanings:

(i) It can mean finding an antiderivative, or primitive, $F$ of a given function $f$. In this sense integration is just the inverse of differentiation, and there is no theorem whatsoever involved.

(ii) Integration can mean the mental process of capturing "the area under the curve $y=f(x)$ for $a\leq x\leq b\>$" as a limit of Riemann sums. The FTC tells us that this limit is equal to $F(b)-F(a)$, so that it can be magically computed by clever handling of function terms.

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