How to integrate $e^{x^2}$?

Question

I am stuck in this problem of integrating $e^{x^2}$. I was solving the linear differential equation of second order for damped oscillations in which i got this to solve

Answer 1

The integral of the function $f(x)= e^{x^2}$ cannot be expressed in terms of elementary functions.

The integral can be given in terms of the imaginary error function, $\text{erfi}(x)$, which is defined as: $$\text{erfi}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{t^2} dt $$