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

Existence of Integral
Though this integral doesn't show that $\zeta\left({2}\right) = \frac {\pi^2} 6$, it shows an interesting relationship between the Riemann Zeta function and the natural logarithm.

Consider:


 * $\displaystyle \int_{\to 0}^{\to 1} \ln x \ln \left({1-x}\right)\ \mathrm dx$

To prove the integral exists, note that $\ln$ is continuous for all $x$ in its domain, in particular $\forall x \in \left({0..1}\right)$.

For $x \to 0^+$:

For $x \to 1^{-}$: