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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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 $y=\dfrac{1}{x}$ – FDP May 03 '16 at 11:11
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Thank you! this makes it clear. let me try to work it out. I think this is enough for an answer. – user6043040 May 03 '16 at 11:52
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http://math.stackexchange.com/questions/850442/an-interesting-identity-involving-powers-of-pi-and-alternating-zeta-series – Jack D'Aurizio May 03 '16 at 13:31
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thank you @JackD'Aurizio, I find this post is also related and useful! Your answer below is exactly the same thing as my post?! – user6043040 May 04 '16 at 05:57