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

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

Consider:


 * $\ds \int_{\to 0}^{\to 1} \ln x \map \ln {1 - x} \rd x$

To prove the integral exists, note that $\ln$ is continuous for all $x$ in its domain, in particular $\forall x \in \openint 0 1$.

For $x \to 0^+$:

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