Could anybody know formulation of integration
$\displaystyle f(x)=\int _0 ^x e^{-t^2}dt$
Formulation is not series.
It doesn't. Read this: Integration of Nonelementary Functions
If k = 1.24 , then
$\displaystyle \int _0 ^x e^{-t^2}dt \approx \frac{\sqrt{\pi (1-e^{-kx^2})}}{2}$
It will approach very small error.
Could you find k or k(x) to minimum error ?
If we let limit of error to 0, it mean we get formulation
As your claim, "There are two things you should never try to prove ...... the impossible and the obvious."
You never prove it doesn't.