What is the closed form of $\int e^{x^2} \, dx$?

How can I do a closed form expansion of $\int e^{x^2} \, dx$? Please be specific as to which method I must use.

Solution 1:

$\mbox{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{-\infty}^x e^{-t^2} \ dt$ is known (at least after multiplication by a constant) as the Error function. There is no simple closed form for the Error function in terms of elementary functions such as polynomial, exponential or trigonometric functions.

Through a little manipulation, you can show that $\frac{2}{\sqrt{\pi}}\int_{0}^x e^{t^2} \ dt = i\mbox{ erf}(ix)$, which is known as the imaginary Error function. Clearly, since it's just a transformation of the normal Error function, it has no nice closed form either.