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Lets say we have a function $f:\mathbb{R} \to \mathbb{R}$. How is it possible to tell that we cannot find the anti-derivative of the function? Is there any specific test for it?

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    $\begingroup$ Do you mean anti-derivative or anti-derivative in terms of elementary functions? $\endgroup$
    – mvw
    Commented Sep 14, 2016 at 10:38
  • $\begingroup$ I think that you want this. $\endgroup$
    – Ivo Terek
    Commented Sep 14, 2016 at 10:42
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    $\begingroup$ As a rule of thumb, functions have no closed-form antiderivative. :-) $\endgroup$
    – user65203
    Commented Sep 14, 2016 at 10:49
  • $\begingroup$ If you mean in terms of elementary functions, see here: math.stackexchange.com/questions/265780/… $\endgroup$ Commented Sep 14, 2016 at 12:47

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If you mean anti-derivatives in general, there seems to be no simple test.

However mathematicians have proven that certain classes of functions have an anti-derivative. E.g. a continous function has an anti-derivative. So all tests that prove a function to be continous would apply.

For the more weird cases see here.

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  • $\begingroup$ how about exp(x^2) ? that can be proved to be not integrable without the error function, which is defined in terms of the integral of exp(x^2) - if a function is continuous and continuously differentiable, then it is always possible to integrate its Taylor Series expansion - then call that function $\endgroup$
    – Cato
    Commented Sep 14, 2016 at 10:53
  • $\begingroup$ You mean $e^{-x^2}$. Has an antiderivative. That it is defined in terms of an integral expression or a special function is just a matter of convenience. $\endgroup$
    – mvw
    Commented Sep 14, 2016 at 10:55
  • $\begingroup$ what is the antiderivative of $e^{-x^2}$ or $e^{x^2}$? Since they are surely continuous? $\endgroup$
    – Cato
    Commented Sep 14, 2016 at 11:09
  • $\begingroup$ @AndrewDeighton $f(x)=\int_0^x e^{-t^2}\,dt$ has the property that $f'(x)=e^{-x^2}$. See math.stackexchange.com/questions/523824/… $\endgroup$
    – egreg
    Commented Sep 14, 2016 at 11:33

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