Calculate $\lim _{n\to \infty }\left(n\int _0^1x^n\sqrt{1+x^2}\,{\rm d}x\right)$

Question

It can be solved by using First mean value theorem for definite integrals, but I have no idea about the exact answer, only have a answer range. There is a similar question: $\lim _{n\to \infty }\left(\int _0^1x^n\sqrt{1+x^2}\right)$ By using First mean value theorem for definite integrals, it turns into: $\lim _{n\to \infty }\sqrt{1+ξ^2}\int _0^1\:x^ndx$ Because $\int _0^1\:x^ndx=\frac{1}{n+1}$→0, $1<\sqrt{1+ξ^2}<\sqrt{2}$ so the original formula=0.