\(\displaystyle \zeta (2) = \sum^{\infty}_{n=1} \frac{1}{n^{2}} = Li_{2} (1) = -\int^{1}_{0} \frac{\ln(1-x)}{x} \ dx \)

So how do I show that \(\displaystyle -\int^{1}_{0} \frac{\ln(1-x)}{x} \ dx = \frac{\pi^{2}}{6} \) ?

The only solution I've found is to write it as \(\displaystyle \int^{1}_{0} \int^{1}_{0} \frac{1}{1+xy} \ dy \ dx \) and then make a crazy change of variables that rotates the coordinate system by \(\displaystyle \frac{\pi}{4}\). But that gets really messy.

So how do I show that \(\displaystyle -\int^{1}_{0} \frac{\ln(1-x)}{x} \ dx = \frac{\pi^{2}}{6} \) ?

The only solution I've found is to write it as \(\displaystyle \int^{1}_{0} \int^{1}_{0} \frac{1}{1+xy} \ dy \ dx \) and then make a crazy change of variables that rotates the coordinate system by \(\displaystyle \frac{\pi}{4}\). But that gets really messy.

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