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Question Number 116996 by mnjuly1970 last updated on 08/Oct/20

Answered by mindispower last updated on 09/Oct/20

$$thelastis{li}_3(\frac{1}{x}-1)\int ?$$

Commented by mnjuly1970 last updated on 09/Oct/20

$$no$$$${li}_3(1-\frac{1}{x})$$

Commented by mnjuly1970 last updated on 09/Oct/20

$${it}^{\prime }sproofisverynice.$$

Answered by mindispower last updated on 19/Oct/20

$$\frac{d}{dx}{Li}_{s+1}\left(x\right)=\frac{{li}_s\left(x\right)}{x}$$$$f\left(x\right)={li}_3\left(x\right)+{li}_3(1-x)+{li}_3(1-\frac{1}{x})$$$$f^{\prime }\left(x\right)=\frac{{li}_2\left(x\right)}{x}-\frac{{li}_2(1-x)}{1-x}+\frac{1}{x^2}{Li}_2(1-\frac{1}{x}).\frac{x}{x-1}$$$$=\frac{(1-x){li}_2\left(x\right)-{xli}_2(1-x)-{li}_2(1-\frac{1}{x})}{x(1-x)}$$$${Li}_2\left(x\right)+{li}_2(1-x)=\frac{{\pi }^2}{6}-ln\left(x\right)ln(1-x)..1$$$${li}_2(1-x)+{li}_2(1-\frac{1}{x})=-\frac{{ln}^2\left(x\right)}{2}..2$$$$f^{\prime }\left(x\right)=\frac{(1-x)({li}_2\left(x\right)+{li}_2(1-x))-{\mathrm{li}}_2(1-\mathrm{x})-{\mathrm{li}}_2(1-\frac{1}{\mathrm{x}})}{\mathrm{x}(1-\mathrm{x})}$$$$=\frac{{\pi }^2}{6x}+\frac{{ln}^2\left(x\right)}{2x(1-x)}$$$$f\left(x\right)=\frac{{\pi }^2}{6}ln\left(x\right)+\int {ln}^2\left(x\right)\left(\frac{x+(1-x)}{2x(1-x)}\right)dx$$$$=\frac{{\pi }^2}{6}ln\left(x\right)+\frac{1}{2}\int \frac{{ln}^2\left(x\right)}{x}dx+\int \frac{{ln}^2\left(x\right)}{2(1-x)}d\underset{=A}{x}$$$$Abypart$$$$=\frac{{\pi }^2}{6}ln\left(x\right)+\frac{1}{6}{ln}^3\left(x\right)+-\frac{ln(1-x){ln}^2\left(x\right)}{2}+/\int ln\left(x\right)ln(1-x).\frac{1}{x}\underset{missingone}{/}$$$$f\left(x\right)={Li}_3\left(x\right)+{Li}_3(1-\frac{1}{x})+{li}_3(1-x)$$$$=\frac{{\pi }^2}{6}ln\left(x\right)+\frac{{ln}^3\left(x\right)}{6}-\frac{{ln}^2\left(x\right)ln(1-x)}{2}+c$$$$\underset{x\to 1}{\lim }f\left(x\right)={Li}_3\left(1\right)=\sum _{n⩾1}\frac{1}{n^3}=\zeta \left(3\right)=c$$$$\Rightarrow {Li}_3\left(x\right)+{li}_3(1-x)+{li}_3(1-\frac{1}{x})$$$$=\zeta \left(3\right)+\frac{{\pi }^2}{6}ln\left(x\right)+\frac{{ln}^3\left(x\right)}{6}-\frac{{ln}^2\left(x\right)ln(1-x)}{2}whatwewant$$$$toobeecontibuedtheAintegralmissing$$$$$$

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