[tex]e^{-x^2}[/tex] has no antiderivative in terms of elementary functions (polynomials, exponentials, logarithms, trigonometric functions, etc), but there is a special function defined to fit that role called the error function, [tex]\mathrm{erf}(x)[/tex], where
[tex]\mathrm{erf}(x)=\displaystyle\frac2{\sqrt\pi}\int_0^xe^{-t^2}\,\mathrm dt[/tex]
By the fundamental theorem of calculus, we can see that
[tex]\dfrac{\mathrm d}{\mathrm dx}\mathrm{erf}(x)=\dfrac2{\sqrt\pi}e^{-x^2}[/tex]
which means we have
[tex]\displaystyle\int e^{-x^2}\,\mathrm dx=\dfrac{\sqrt\pi}2\mathrm{erf}(x)+C[/tex]
