So a basic form of the Gauss error function is
$$\int e^{x^2}\,dx$$
and apparently this is not solvable analytically. But why? It seems that I can solve it pretty easily as
$$\int e^{x^2}\,dx = \frac{1}{2x}e^{x^2}$$
since
$$\frac{d}{dx} \frac{1}{2x}e^{x^2} = e^{x^2}.$$
Why is this wrong?
When you take the derivative of $$\frac{1}{2x}e^{x^{2}}$$ you need to use the product and chain rules. You get $$\frac{d}{dx}\frac{1}{2x}e^{x^{2}} = -\frac{1}{2x^{2}}e^{x^{2}}+\frac{1}{2x}2xe^{x^{2}} = e^{x^{2}}\left(1-\frac{1}{2x^{2}}\right) \neq e^{x^{2}}. $$