In calculus I, I encountered the function $e^{-x^2}$. The teacher told us that it was impossible to integrate. Curious, I graphed the function on my calculator, together with its integral. The integral appeared to be a perfectly ordinary function. I wondered whether this function's integral had any special properties as a function, apart from being impossible to define in terms of elementary functions.
Now, I have encountered many such impossible to integrate functions. Are there any such functions that have special properties apart from being impossible to integrate?
They are more or less ordinary functions. For example, consider the function $$f(x)=e^{x^2}\int_0^x{e^{-t^2}\,\mathrm{d}t}$$ Then $$f'(x)=2xe^{x^2}\int_0^x{e^{-t^2}\,\mathrm{d}t}+1$$ So $$f'(x)=2xf(x)+1$$ In other words, this functions is a solution to the differential equation $y'-2xy=1$. The function you mentioned also has the remarkable property that $$\int_{-\infty}^{\infty}{e^{-x^2}\,\mathrm{d}x}=\sqrt \pi$$ Aside from that, there is nothing remarkable per se about functions that cannot be expressed in terms of elementary functions. Many power series, for example, have the same property.