What does elementary function mean? I am looking at my double integration example problems and I see a note that says integral of $e^{-x^2}$ is not an elementary function. What does that mean? And why isn't that an elementary function?
It actually says observe integral of $e^{-x^2} dx$ is not an elementary function. But why is that? Same goes for integral of (sin(x)/x) dx, why isn't that an elementary function also?
 A: The elementary functions form the smallest differential field that contains the complex constants and 
the variable $x$ and such that:


*

*If $f' = g'/g$ where $g$ is elementary then $f$ is elementary (i.e. $\log(g)$ is elementary)

*If $f' = g' f$ where $g$ is elementary then $f$ is elementary (i.e. $\exp(g)$ is elementary)

*If $P(f) = 0$ where $P$ is a nonconstant polynomial with elementary coefficients then $f$ is elementary.


The theory of integration in elementary functions dates back to Liouville in 1833. 
For more, see e.g. this sci.math article
A: An elementary function is a function built from any combination of $+,-, \cdot, /$ and function composition with polynomial functions, exponential functions, logarithms, and the trigonometric functions of one variable. 
It is very likely they meant to say the error function 
$$
\mathrm{erf}(x) = \frac{2}{\pi} \int_0^x e^{-t^2}\ dx.
$$
The proof that this function is not elementary (the proof would essentially show that one cannot represent the integral as a finite amount of elementary functions) is quite involved. The author of the book is likely trying to demonstrate the integration of an elementary function ($e^{-t^2}$ is elementary) does not necessarily yield and elementary function as the result. One more time,
$e^{-t^2}$ is elementary but $\int e^{-t^2}$ is not elementary.
