That cannot be integrated in terms of "elementary" functions. If we let u= ix, then du= idx and \(\displaystyle x^2= -u^2\) so that
\(\displaystyle \int e^{x^2} dx= i\int e^{-u^2} du= i \frac{\sqrt{2}}{\pi} erf(u)= i erf(ix)\) where "erf" is the "error function", defined by \(\displaystyle erf(x)= \frac{\pi}{\sqrt{2}}\int_0^x e^{-t^2}dt\). That integral also cannot be done in terms of "elementary" functions but has been tabulated and included on some calculators. It is called the "error function" because the function \(\displaystyle e^{-x^2}\) arises in the "normal probability distribution" that itself occurs in calculations of errors in measurement.
