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How do I prove that $$\int_{0}^{1/2} \dfrac{\log (1-x)}{x} \mathrm{d}x=\dfrac{1}{2}\log^2{2}-\dfrac{\pi^2}{12}$$

I taylor expanded $\log(1-x)$ but ended up with a series I couldn't evaluate. Please help. Thank you.

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  • $\begingroup$ I don't think so that 1÷2(log2)^2 would be present. $\endgroup$ – Aditya Kumar Mar 8 '15 at 7:55
  • $\begingroup$ @AdityaKumar yes it would be $\endgroup$ – RE60K Mar 8 '15 at 7:56
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$$\frac{\log(1-x)}x=-\frac{(x+x^2/2+x^3/3+x^4/4+...)}x=-(1+x/2+x^2/3+x^3/4+...)$$ Now: $$\int\frac{\log(1-x)}x{\rm d}x=-(x+x^2/2^2+x^3/3^3+...)=-{\rm Li}_2(x)$$ Now: $$\int_0^{1/2}\frac{\log(1-x)}x{\rm d}x={\rm Li}_2(0)-{\rm Li}_2(1/2)=\frac{\log^22}2-\frac{\pi^2}{12}$$ since ${\rm Li}_2(0)=0$ and by duplication formula: $$\mathrm{Li}_2(x)+\mathrm{Li}_2(1-x) =\frac{\pi^2}{6}-\log(x)\log(1-x)$$ Put $x=1/2$: $$2\mathrm{Li}_2(1/2)=\frac{\pi^2}{6}-\log(1/2)\log(1/2)\implies \mathrm{Li}_2(1/2)=\frac{\pi^2}{12}-\frac12\log^2(2)$$

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  • $\begingroup$ I suppose that $x^3/3^3$ would be $x^3/3^2$ $\endgroup$ – Claude Leibovici Mar 8 '15 at 8:36
  • $\begingroup$ @ClaudeLeibovici you know... $\endgroup$ – RE60K Mar 9 '15 at 9:15

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