I've seen this many times: someone will post a homework question asking them to prove something about an integral [tex] \int_a^b f(t) dt [/tex], where f is some arbitrary, continuous, integrable function. They will write in their proof, [tex] \int_a^b f(t) dt = F(b) - F(a) [/tex] where they're assuming F is the anti-derivative of f. They will get told that they cannot assume that f has an anti-derivative.(adsbygoogle = window.adsbygoogle || []).push({});

My question now is, if f is continuous and integrable, then doesn't it follow from the FTC that there exists a function F such that F' = f? You may not be able to express F in terms of elementary functions, but F still must exist, right? So why can't the student write in their proof [tex] \int_a^b f(t) dt = F(b) - F(a) [/tex]???

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# Anti-derivatives and Elementary Functions

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