# How to know if the function is non integrable?

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?

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

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.

• 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
– Cato
Sep 14, 2016 at 10:53
• 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.
– mvw
Sep 14, 2016 at 10:55
• what is the antiderivative of $e^{-x^2}$ or $e^{x^2}$? Since they are surely continuous?
– Cato
Sep 14, 2016 at 11:09
• @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/… Sep 14, 2016 at 11:33