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I've been reading through Stewart's calculus, and it mentions how most elementary functions don't have elementary antiderivatives. I don't understand why the derivative of an elementary function is an elementary function, but the inverse operation(antiderivative) isn't. Below is a snippet of the text,

For instance, the function $$f(x)=\sqrt{\frac{x^2-1}{x^3+2x-1}}+\ln(\cosh(x))-xe^{\sin(2x)}$$ is an elementary function.

If $f$ is an elementary function, then $f’$ is an elementary function but $\int f(x)dx$ need not be an elementary function. Consider $f(x)=e^{x^2}$. Since $f$ is continuous, its integral exists, $\ldots$

Colton Fitzjarrald
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    It's just how inverses tend to work in mathematics, intuitively speaking. The sum of two positive integers is a positive integer, but that need not apply to differences. The product of two positive integers is a positive integer, but that need not apply to quotients. The exponentiation of positive integers is a positive integer, but that need not apply to roots or logarithms. While it doesn't really "answer" the question, I think the gist that "inverses of nice things tend to be dirty" is a pattern you see a lot and worth noting here, and that this is an extension of that. – PrincessEev Oct 17 '22 at 17:18
  • Related questions. – J.G. Oct 17 '22 at 17:23
  • @PrincessEev I agree with the sentiment of your comment, but to apply it here would be to assume that differentiation is the "forward" operation, while taking the anti-derivative would be the (difficult) "backward" operation. Now, I do think that there is a fair case to be made for it being that way around (since anti-differentiation is only supposed to undo a differentiation), but I don't know if it's as obvious in that case as with the other examples of operations. – Jam Oct 17 '22 at 17:44
  • More related questions: Q2194769 Q2199236 – Jam Oct 17 '22 at 17:46
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    Is there any reason to expect the antiderivative of an elementary function to be elementary? – R. Burton Oct 17 '22 at 18:27

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