Proving a limit of an integral I don't know how to approach this problem.
Proof that $\lim_{n\to \infty} (n+1)I_n = \frac 12$
where $I_n = \int_{0}^{1} \frac {x^n}{x+1}dx$ 

Hint:
$$(n+1)\int_0^1 \frac{x^n}{1+x}\, dx - \frac{1}{2} = (n+1)\int_0^1  x^n\left ( \frac{1}{1+x} - \frac{1}{2}\right )\,dx.$$
