I would like to find a simple equivalent of:
$ u_{n}=\frac{1}{n!}\int_0^1 (\arcsin x)^n \mathrm dx $
We have:
$ 0\leq u_{n}\leq \frac{1}{n!}\left(\frac{\pi}{2}\right)^n \rightarrow0$
So $ u_{n} \rightarrow 0$
Clearly:
$ u_{n} \sim \frac{1}{n!} \int_{\sin(1)}^1 (\arcsin x)^n \mathrm dx $
But is there a simpler equivalent for $u_{n}$?
Using integration by part:
$ \int_0^1 (\arcsin x)^n \mathrm dx = \left(\frac{\pi}{2}\right)^n - n\int_0^1 \frac{x(\arcsin x)^{n-1}}{\sqrt{1-x^2}} \mathrm dx$
But the relation
$ u_{n} \sim \frac{1}{n!} \left(\frac{\pi}{2}\right)^n$
seems to be wrong...