I would like to get an asymptotic limit at the following integral: for $p\ge 2, n \in N$, $t \ge 0$ $ \int_{0}^{\frac 12 \sqrt{(n+1)!}}\left(1-\frac{t^2}{2^2(n+1)!}\right)^p \mathrm{d} t $ I think substitution $t=\frac 12 \sqrt{(n+1)!}y$ should work. But after the substittution, I don't know what to do.
Thank you for your help.