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Question Number 137610 by mnjuly1970 last updated on 04/Apr/21

          ........ mathematical   analysis (II)....      prove  that ::       𝛀=∫_0 ^( 1) (1/(1+x))ln(((x^2 +2x+1)/(1+x+x^2 )))=Σ_(n=1) ^∞ (1/(n^2  (((2n)),((  n)) )))=(π^2 /(18))..

$$\:\:\:\:\:\:\:\:\:\:........\:{mathematical}\:\:\:{analysis}\:\left({II}\right).... \\ $$$$\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\boldsymbol{\Omega}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}}{\mathrm{1}+{x}}{ln}\left(\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} \begin{pmatrix}{\mathrm{2}{n}}\\{\:\:{n}}\end{pmatrix}}=\frac{\pi^{\mathrm{2}} }{\mathrm{18}}.. \\ $$

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