>>212
Well, there is certainly an antiderivative function for \[f(x)=e^{-x^2}\] and you can plot it (pic rel), it just can't be written in terms of elementary functions.  \href{https://en.wikipedia.org/wiki/Elementary_function}{Elementary function} is any finite composition of addition, multiplication, polynomials (\[1,x^2,x^3,\dots\]), trigonometric and inverse trigonometric functions (\[\sin(x), \cos(x), \tan(x),\dots\], \[\arcsin(x),\arccos(x),\arctan(x),\dots\]), powers (\[x^a\]), exponentials (\[e^x\]) and logarithms (\[\log(x)\]) i.e. everything you learn in high school.

There are plenty of functions whose antiderivatives can't be written in terms of elementary functions.

>Can we define a function to be the solution 
Yes. Since you can't write the solution \[e^{-x^2}\] in terms of elementary functions, the integral called "unsolvable", but you can define a new function to say "this function is the solution to this integral". And in fact, that's the error function.

,,\qquad \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^xe^{-x^2}\mathrm{d}x

There are many other functions defined this way (e.g. the gamma function \[\Gamma(x) = \int_0^{\infty}x^{n-1}e^{-x}\mathrm dx\]) and they're used daily, but they are not considered elementary functions in the usual sense and neither Bessel functions \[J_\alpha(z),Y_\alpha(z)\], Lambert \[W(z)\] function, sine and cosine integral functions \[\mathrm{Si}(x), \mathrm{Ci}(x)\], Fresnel integral functions \[S(x), C(x)\] etc.