Basel-related integrals

Given that:

\displaystyle \sum_{n = 1}^\infty \frac 1 {n^2} = \frac {\pi^2} 6

Show that:

\displaystyle \sum_{n = 1}^\infty \frac {(-1)^{n - 1}} {n^2} = \frac {\pi^2} {12}

(i) Show that:

\displaystyle \int_0^1 \log x \log (1 - x) \mathrm dx = 2 - \frac {\pi^2} 6

(ii) Evaluate:

\displaystyle \int_0^1 \log x \log (1 + x) \mathrm dx

You can ignore issues of convergence and swapping sums/integrals.

Not too hard - just neat.

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Minor error at the end of the second, you wrote 1 + 1 = 1, but otherwise good work.

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