Question Number 132469 by Algoritm last updated on 14/Feb/21
Answered by Olaf last updated on 15/Feb/21
![Let Ω_i = ∫_0 ^1 (√(x_i (1−lnx_i )))dx_i Let u_i = 1−lnx_i Ω_i = ∫_(+∞) ^1 (√(e^(1−u_i ) u_i ))(−e^(1−u_i ) du_i ) Ω_i = ∫_1 ^(+∞) e^((3/2)(1−u_i )) u_i ^(1/2) du_i Ω_i = e^(3/2) ∫_1 ^(+∞) e^(−(3/2)u_i ) u_i ^(1/2) du_i Ω_i = e^(3/2) [((√(6π))/9)(1−erf(((√6)/2))+(√(6/π))e^(−(3/2)) )] Ω_i ≈ 0∙84668... Ω = ∫_0 ^1 ∫_0 ^1 ...∫_0 ^1 Π_(i=1) ^n (√(x_i (1−lnx_i )))dx_i Ω = Π_(i=1) ^n ∫_0 ^1 (√(x_i (1−lnx_i )))dx_i Ω = Π_(i=1) ^n Ω_i = (0.84668...)^n](https://www.tinkutara.com/question/Q132556.png)
$$\mathrm{Let}\:\Omega_{{i}} \:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{{x}_{{i}} \left(\mathrm{1}−\mathrm{ln}{x}_{{i}} \right)}{dx}_{{i}} \\ $$$$\mathrm{Let}\:{u}_{{i}} \:=\:\mathrm{1}−\mathrm{ln}{x}_{{i}} \\ $$$$\Omega_{{i}} \:=\:\int_{+\infty} ^{\mathrm{1}} \sqrt{{e}^{\mathrm{1}−{u}_{{i}} } {u}_{{i}} }\left(−{e}^{\mathrm{1}−{u}_{{i}} } {du}_{{i}} \right) \\ $$$$\Omega_{{i}} \:=\:\int_{\mathrm{1}} ^{+\infty} {e}^{\frac{\mathrm{3}}{\mathrm{2}}\left(\mathrm{1}−{u}_{{i}} \right)} {u}_{{i}} ^{\mathrm{1}/\mathrm{2}} {du}_{{i}} \\ $$$$\Omega_{{i}} \:=\:{e}^{\frac{\mathrm{3}}{\mathrm{2}}} \int_{\mathrm{1}} ^{+\infty} {e}^{−\frac{\mathrm{3}}{\mathrm{2}}{u}_{{i}} } {u}_{{i}} ^{\mathrm{1}/\mathrm{2}} {du}_{{i}} \\ $$$$\Omega_{{i}} \:=\:{e}^{\frac{\mathrm{3}}{\mathrm{2}}} \left[\frac{\sqrt{\mathrm{6}\pi}}{\mathrm{9}}\left(\mathrm{1}−\mathrm{erf}\left(\frac{\sqrt{\mathrm{6}}}{\mathrm{2}}\right)+\sqrt{\frac{\mathrm{6}}{\pi}}{e}^{−\frac{\mathrm{3}}{\mathrm{2}}} \right)\right] \\ $$$$\Omega_{{i}} \:\approx\:\mathrm{0}\centerdot\mathrm{84668}… \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} …\int_{\mathrm{0}} ^{\mathrm{1}} \underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\sqrt{{x}_{{i}} \left(\mathrm{1}−\mathrm{ln}{x}_{{i}} \right)}{dx}_{{i}} \\ $$$$\Omega\:=\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{{x}_{{i}} \left(\mathrm{1}−\mathrm{ln}{x}_{{i}} \right)}{dx}_{{i}} \\ $$$$\Omega\:=\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\prod}}\Omega_{{i}} \:=\:\left(\mathrm{0}.\mathrm{84668}…\right)^{{n}} \\ $$