User:GFauxPas/Sandbox/Zeta2/lnxln1-x/evaluation

Evaluation of Integral
Let $n \in \N_{\ge 1}$

So,

Since $n$ was arbitrary, this holds for all cases $n \ge 1$, for $x \in (0\,.\,.\,1)$.

Since we assumed uniform continuity:

For $x \to 0^+$:

This means that the value of the integral in question depends only on the limit as $x \to 1^{-}$:

Using partial fraction decomposition:

Clearly $A = 1, C = -1, B = -1$.

The Riemann Zeta function at $2$ is famously equal to $\dfrac {\pi^2}6$.

That is,