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I need hints on the proof of:

$$\int_0^\infty\dfrac{\ln(x)^2}{1+x^2}{\rm{d}x}=\dfrac{\pi^3}{8}$$

and then:

$$\sum\limits_{n=0}^\infty\left((-1)^n\dfrac{1}{(2n+1)^3}\right)=\dfrac{\pi^3}{32}$$

Thank you very much!

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  • $\begingroup$ Check out the value of $\displaystyle \int_0^1 x^n(\ln x)^2 dx$ and the Taylor development of $\dfrac{1}{1+x^2}$ and think about change of variable $y=\dfrac{1}{x}$ $\endgroup$ – FDP May 3 '16 at 11:11
  • $\begingroup$ Thank you! this makes it clear. let me try to work it out. I think this is enough for an answer. $\endgroup$ – user6043040 May 3 '16 at 11:52
  • $\begingroup$ math.stackexchange.com/questions/850442/… $\endgroup$ – Jack D'Aurizio May 3 '16 at 13:31
  • $\begingroup$ thank you @JackD'Aurizio, I find this post is also related and useful! Your answer below is exactly the same thing as my post?! $\endgroup$ – user6043040 May 4 '16 at 5:57

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