Why can't $e^{x^2}$ be integrated My teacher told me that not only do we have to use the erf function to approximate error, but that it is proved impossible to integrate in real analysis (at least not Riemann-integrable). Is there a name for this proof, and can I have it? I am not a mathematician, but such a simple function has been around and it irks me terribly and need a detailed proof for closure.
 A: Your teacher probably said that $\exp(-x^2)$ (or $\exp(x^2)$) does not have an antiderivative that can be expressed using elementary functions:

In mathematics, an elementary function is a function of one variable built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations (+ – × ÷).

The set of elementary functions forms a field of functions, equipped with a derivative operation: a differential field of functions. These kind of fields have been introduced by Liouville, in order to demonstrate his famous theorem which states that 

the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.

The proof uses algebra in the differential field of functions. It would be by all means not productive to reproduce it here, but there are some references you might want to check:


*

*The modern version of the demonstration of this theorem is an open-access article by Rosenlicht.

*A post by Matthew Wiener provides several examples.

A: As V. Rossetto wrote, your teacher more than likely told that the antiderivative of such a function cannot be obtained on the basis of simple functions.  
For this integrand, the antiderivative is  Sqrt[Pi] Erfi[x] / 2, where Erfi[x]is the imaginary error function Erf(iz)/i. The error function Erf[x] is the integral of the Gaussian distribution, given by Erf[z]= (2 /Sqrt[Pi]) Integral[Exp[-t^2],{t,0,z}].   
I hope that you better percieve the vicious circle we move around. 
