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How to calculate the following integral?

$$\int_0^1\frac{\ln x}{x^2-x-1}\mathrm{d}x=\frac{\pi^2}{5\sqrt{5}}$$

Jack D'Aurizio
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Born to be proud
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    Please show your own work, and where you got the problem from, some people on this site respond better to well thought out and motivated questions... – Zach466920 Jul 30 '15 at 02:00
  • It is an abstract duplicate of this question: http://math.stackexchange.com/questions/1374834/some-infinite-series-with-fibonacci-numbers?lq=1 – Jack D'Aurizio Jul 30 '15 at 03:15

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You can start writing $$x^2-x-1=(x-r_1)(x-r_2)$$ where $r_{1,2}=\frac{1}{2} \left(1\pm\sqrt{5}\right)$ and use partiel fraction decomposition. So,$$\frac 1{x^2-x-1}=\frac{1}{{r_2}-{r_1}} \Big(\frac{1}{x-r_2}-\frac{1}{x-r_1}\Big)$$ and use $$\int \frac{\log(x)}{x+a}=\text{Li}_2\left(-\frac{x}{a}\right)+\log (x) \log \left(1+\frac{x}{a}\right)$$

Claude Leibovici
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