Question

How do I find the antiderivative of e^-x^2?

Answer 1

You can't find one in terms of elementary functions, but you can express it in a variety of ways.

Consider the power series method.
\[e^x=\sum_{k=0}^\infty \frac{x^k}{k!}~~\implies~~e^{-x^2}=\sum_{k=0}^\infty \frac{(-x^2)^k}{k!}=\sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{k!}\]
Integrating with respect to \(x\) gives
\[\int e^{-x^2}\,dx=\sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{k!(2k+1)}\]
Another way, using the fundamental theorem of calculus and what's known as the error function:
\[e^{-x^2}=\frac{d}{dx}\int_0^x e^{-t^2}\,dt=\frac{d}{dx}\left[\frac{\sqrt\pi}{2}\text{erf}(x)\right]\]
See here: http://en.wikipedia.org/wiki/Error_function