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Prove-that-1-ln-x-x-pi-1-ln-2-x-1-2-dx-pi-2-8-16-




Question Number 153875 by mnjuly1970 last updated on 11/Sep/21
      Prove  that..           𝛗 : =∫_( 1) ^( +∞) (( ln (x ))/(( x^( π)  −1 )( ln^( 2) (x) +1 )^2 ))dx= ((π^( 2) − 8)/(16))         ■
$$ \\ $$$$\:\:\:\:\mathrm{Prove}\:\:\mathrm{that}.. \\ $$$$\:\:\: \\ $$$$\:\:\:\:\boldsymbol{\phi}\::\:=\int_{\:\mathrm{1}} ^{\:+\infty} \frac{\:{ln}\:\left({x}\:\right)}{\left(\:{x}^{\:\pi} \:−\mathrm{1}\:\right)\left(\:{ln}^{\:\mathrm{2}} \left({x}\right)\:+\mathrm{1}\:\right)^{\mathrm{2}} }{dx}=\:\frac{\pi^{\:\mathrm{2}} −\:\mathrm{8}}{\mathrm{16}}\:\:\:\:\:\:\:\:\:\blacksquare\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

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