I ran across what appears to be another Gamma identity.
Show that $$\lim_{n\to \infty}n^{p+1}\int_{0}^{1}e^{-nx}\ln(1+x^{p}) \,\mathrm dx=\Gamma(p+1)=p!$$
I tried several different subs and even the series for $$\ln(1+x^{p})$$ and nothing materialized.
What would be a good start on this one?
It looks similar to $$\Gamma(p+1)=n^{p+1}\int_{0}^{\infty}x^{p}e^{-nx} \,\mathrm dx,$$ but I was unable to hammer it into that form. I would think there is a clever sub of some sort that may work it into this last mentioned form.