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Question Number 163751 by amin96 last updated on 10/Jan/22

$${\int }_0^1\mathit{l}\mathit{n}(1+\mathit{x})\mathit{l}\mathit{n}(1-\mathit{x})\mathit{d}\mathit{x}=?$$$$\mathit{b}\mathit{y}\mathit{M}\mathit{A}\mathit{T}\mathit{H}.\mathit{A}\mathit{M}\mathit{I}\mathit{N}$$$$--------------------$$

Answered by Kamel last updated on 10/Jan/22

$$\mathrm{\Omega }=-\frac{1}{2}({\int }_0^1{Ln}^2\left(\frac{1-x}{1+x}\right)dx-{\int }_0^1{Ln}^2(1-x)dx-{\int }_0^1{Ln}^2(1+x)dx)$$$$=-\frac{1}{2}(2{\int }_0^1\frac{{Ln}^2\left(t\right)}{{(1+t)}^2}dt-2-2{Ln}^2\left(2\right)+2{\int }_0^{1}Ln(1+x)dx)$$$$=-\frac{1}{2}(-4{\int }_0^1\frac{Ln\left(t\right)}{1+t}dg-2-2{Ln}^2\left(2\right)+4Ln\left(2\right)-2)$$$$=2+{Ln}^2\left(2\right)-2Ln\left(2\right)-\frac{{\pi }^2}{6}$$

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