Topsquark is correct that $\displaystyle \displaystyle \begin{align*} \int{e^{x^2}\,dx} \end{align*}$ can not be expressed in a closed form in terms of the elementary functions. However, it can be expressed as an infinite series.
$\displaystyle \displaystyle \begin{align*} e^X &= \sum_{n = 0}^{\infty}{\frac{X^n}{n!}} \\ e^{x^2} &= \sum_{n = 0}^{\infty}{\frac{\left( x^2 \right) ^n }{n!}} \\ &= \sum_{n = 0}^{\infty}{\frac{x^{2n}}{n!}} \\ \int{e^{x^2}\,dx} &= \int{\sum_{n = 0}^{\infty}{\frac{x^{2n}}{n!}}} \\ &= \sum_{n = 0}^{\infty}{\frac{x^{2n + 1}}{\left( 2n + 1 \right) n!}} \end{align*}$