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Question Number 208423 by lepuissantcedricjunior last updated on 15/Jun/24

$$\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{o}\mathit{n}\mathit{s}$$$$\mathit{i}=\int \int {\int }_{[0;1]}\frac{\mathit{d}\mathit{x}\mathit{d}\mathit{y}\mathit{d}\mathit{z}}{1-\mathit{x}\mathit{y}\mathit{z}}$$

Answered by Berbere last updated on 15/Jun/24

$$=\int \int [-\frac{1}{xy}ln(1-xy)]dydx$$$$xy=u\Rightarrow dy=\frac{du}{x}$$$$={\int }_0^1\frac{1}{x}{\int }_0^x\frac{ln(1-u)}{-u}dudx$$$$={\int }_0^1\frac{{Li}_2\left(x\right)}{x}dx={Li}_3\left(1\right)=\zeta \left(3\right)$$$${Li}_{n+1}\left(x\right)={\int }_0^x\frac{{Li}_n\left(z\right)}{z}dz$$$${Li}_2\left(x\right)=-{\int }_0^x\frac{ln(1-t)}{t}dt$$$${Li}_3\left(z\right)=\sum _{n⩾1}\frac{z^n}{n^3};\zeta \left(3\right)=\sum _{n⩾1}\frac{1}{n^3}$$

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