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In many texts and books about calculus we see

There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot be expressed in terms of standard functions.

What functions are called standard functions and why?

M. Vinay
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Dante
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    See What makes elementary functions elementary?. – Dave L. Renfro Jun 11 '14 at 14:08
  • Are standard functions exactly the same elementary functions? – Dante Jun 11 '14 at 14:13
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    There is a technical sense used by those who research this area in which "elementary function" also includes all functions $y$ that can be the solution to some algebraic equation with rational function coefficients and $y$ as the unknown variable in the equation, and this meaning will include some functions that can't be explicitly expressed as a finite combination of (additions, multiplications, compositions) of the kinds of functions you're used to seeing. I suspect, however, that "standard function" for your author doesn't mean this, but rather just means explicitly representable this way. – Dave L. Renfro Jun 11 '14 at 14:21

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