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Prove-that-0-1-ln-3-1-x-x-2-dx-3-4-3-2ln-3-2-Prepared-by-M-N-




Question Number 152839 by mnjuly1970 last updated on 02/Sep/21
     Prove  that :     Ω=∫_0 ^( 1) (( ln^( 3)  (1 + x ))/x^( 2) )dx = (3/4) ζ (3 )− 2ln^( 3) ( 2 )        ■         Prepared by:      M.N
$$ \\ $$$$\:\:\:\mathrm{Prove}\:\:\mathrm{that}\:: \\ $$$$ \\ $$$$\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{ln}^{\:\mathrm{3}} \:\left(\mathrm{1}\:+\:{x}\:\right)}{{x}^{\:\mathrm{2}} }{dx}\:=\:\frac{\mathrm{3}}{\mathrm{4}}\:\zeta\:\left(\mathrm{3}\:\right)−\:\mathrm{2ln}^{\:\mathrm{3}} \left(\:\mathrm{2}\:\right)\:\:\:\:\:\:\:\:\blacksquare \\ $$$$\:\:\:\:\:\:\:\mathrm{Prepared}\:\mathrm{by}:\:\:\:\:\:\:\mathrm{M}.\mathrm{N} \\ $$$$ \\ $$

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