This is probably a really easy question, but I was just trying to work out the Gaussian integral and I must be wrong, but I am not sure where I am wrong...
If $\frac{\delta}{\delta x} e^{-x^2} = -2xe^{-x^2}$ then it should mean that we can take the integral of the same expression by just reversing the process. This implies that to take the integral instead of the derivative, I would divide by the derivative of the exponent...
$$ \int -2xe^{-x^2} = e^{-x^2} \implies \int e^{-x^2} = \frac{e^{-x^2}}{-2x} $$
But this doesn't evaluate to $\sqrt{\pi}$ as the famous result does, so I must have gotten something really wrong somewhere. Where have I gone wrong? Why does this integral need such a special derivation different than a normal integral (as it does in the link above).
I started doing this because I thought I would learn why this integral is special because I thought it would become obvious when I tried to do it, but its not obvious to me