Interesting integral: $\int_0^1{\frac{nx^{n-1}}{x+1}}dx$ Find the value of $$\int_0^1{\frac{nx^{n-1}}{x+1}}dx.$$
I had no luck while integrating it. I also tried differentiating w.r.t n but still couldn't reach anywhere. Need help.

Put $y=1+x$ and the integral becomes
$$ \int_{1}^2 \frac{n(y-1)^{n-1}}{y} \, dy = \int_1^2 \sum_{k=0}^n n\binom{n-1}{k} (-1)^{n-k-1} y^{k-1} \, dy = \left[ n(-1)^{n-1}\log{y} + \sum_{k=1}^n \frac{n}{k} \binom{n-1}{k} (-1)^{n-k-1} y^k \right]_1^2 \\
= n(-1)^{n-1}\log{2} + \sum_{k=1}^n \frac{n}{k} \binom{n-1}{k} (-1)^{n-k-1}(2^k-1). $$
