TITLE: Interesting integral: $\int_0^1{\frac{nx^{n-1}}{x+1}}dx$
QUESTION [8 upvotes]: 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.

REPLY [5 votes]: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
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\begin{align}
\int_{0}^{1}{n\,x^{n - 1} \over x + 1}\,\dd x & =
n\int_{0}^{1}{x^{n - 1} - x^{n} \over 1 - x^{2}}\,\dd x
\,\,\,\stackrel{x^{2}\ \mapsto\ x}{=}\,\,\,
{1 \over 2}\,n\int_{0}^{1}{x^{n/2 - 1} - x^{n/2 - 1/2} \over 1 - x}\,\dd x
\\[5mm] & =
{1 \over 2}\,n\bracks{%
\int_{0}^{1}{1 - x^{n/2 - 1/2} \over 1 - x}\,\dd x -
\int_{0}^{1}{1 - x^{n/2 - 1} \over 1 - x}\,\dd x}
\\[5mm] & =
\bbx{{1 \over 2}\,n\pars{H_{n/2 - 1/2} - H_{n/2 - 1}}}
\end{align}

where $\ds{H_{z}}$ is the Harmonic Number.