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Question Number 142791 by mnjuly1970 last updated on 05/Jun/21

$$Evaluate::...$$$$\mathrm{\Omega }:={\int }_0^1\frac{{li}_2\left(\sqrt{x}\right)}{1+\sqrt{x}}dx=??$$$$...........$$

Answered by Dwaipayan Shikari last updated on 05/Jun/21

$$2{\int }_0^1\frac{{uLi}_2\left(u\right)}{1+u}du$$$$=2{\int }_0^1{Li}_2\left(u\right)du-2{\int }_0^1\frac{{Li}_2\left(u\right)}{1+u}du$$$$=2\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{1}{n^2(n+1)}-\frac{{\pi }^2}{3}log\left(2\right)+\frac{5}{4}\zeta \left(3\right)$$$$=2\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{1}{n^2}-\frac{1}{n(n+1)}-\frac{{\pi }^2}{3}log\left(2\right)+\frac{5}{4}\zeta \left(3\right)$$$$=\frac{{\pi }^2}{3}log\left(\frac{e}{2}\right)-2+\frac{5}{4}\zeta \left(3\right)$$

Commented by mnjuly1970 last updated on 05/Jun/21

$$tashakorandsepasmrpayan...$$

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