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$$I = \int_{0}^{1} \log^2(1-x)\log^2(x) dx$$

http://math.stackexchange.com/questions/959701/evaluate-int1-0-log21-x-log2x-dx

Can you take a look at M.N.C.E.'s method?

I don't understand a few things.

Somehow he makes the relation:

$$\frac{4H_n}{(n+1)(n+2)^3} = \frac{\left( \gamma + \psi(-z) \right)^2}{(z+1)(z+2)^3}$$

How is this established?

And this I don't understand,

**why did he integrate it**,?

And then after he states: "

*At the positive integers,"*what is he doing with the residues. I know the residue theorem etc, but I don't understand what he is exactly doing?

Thanks