Let $$\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int\limits\_{0}^x e^{-t^2}dt$$ be the error function. Then, I have tow questions.

1. For a positive integer $n$, is there a close-form solution of 
$f\_n=\int\_{0}^\infty 1- (\mathrm{erf}(x))^n dx$?
2. Is there a positive $c<\infty$ such that $\lim\limits\_{n\to\infty}\frac{f\_n}{\sqrt{\log n}} =c$.

Mathematica returns $1/\sqrt{\pi}$ when $n=1$ and $\sqrt{2/\pi}$ when $n=2$.