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Question Number 132469 by Algoritm last updated on 14/Feb/21

Answered by Olaf last updated on 15/Feb/21

$$\mathrm{Let}{\mathrm{\Omega }}_i={\int }_0^1\sqrt{x_i(1-\ln x_i)}{dx}_i$$$$\mathrm{Let}{u}_i=1-\ln x_i$$$${\mathrm{\Omega }}_i={\int }_{+\mathrm{\infty }}^1\sqrt{e^{1-u_i}{u}_i}(-e^{1-u_i}{du}_i)$$$${\mathrm{\Omega }}_i={\int }_1^{+\mathrm{\infty }}{e}^{\frac{3}{2}(1-u_i)}{u}_i^{1/2}{du}_i$$$${\mathrm{\Omega }}_i=e^{\frac{3}{2}}{\int }_1^{+\mathrm{\infty }}{e}^{-\frac{3}{2}{u}_i}{u}_i^{1/2}{du}_i$$$${\mathrm{\Omega }}_i=e^{\frac{3}{2}}\left[\frac{\sqrt{6\pi }}{9}(1-\mathrm{erf}\left(\frac{\sqrt{6}}{2}\right)+\sqrt{\frac{6}{\pi }}{e}^{-\frac{3}{2}})\right]$$$${\mathrm{\Omega }}_i\approx 0\cdot 84668...$$$$\mathrm{\Omega }={\int }_0^1{\int }_0^1...{\int }_0^1\underset{i=1}{\prod ^n}\sqrt{x_i(1-\ln x_i)}{dx}_i$$$$\mathrm{\Omega }=\underset{i=1}{\prod ^n}{\int }_0^1\sqrt{x_i(1-\ln x_i)}{dx}_i$$$$\mathrm{\Omega }=\underset{i=1}{\prod ^n}{\mathrm{\Omega }}_i={(0.84668...)}^n$$

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