Integral of e^(-x^2): Does it Have an Indefinite Integral?

[Geoffrey]

I know definite integral exists for this between - infinity to + infinity,

does it have an indefinite integral?

Thanks,
Geoff.

 

[HallsofIvy]

Yes. All continuous functions (and, many functions that are not continuous) have indefinite integrals. But the indefinite integral of that function cannot be written in terms of elementary functions. It can, of course, be written in terms of the "error function", erf(x), because that is defined as $2/\sqrt{\pi}$ times that integral. That is,
$$\int e^{-x^2}dx= \frac{\sqrt{\pi}}{2}erf(x)+ C$$