Question Number 219577 by SdC355 last updated on 29/Apr/25
$$\mathrm{prove} \\ $$$$\underset{{n}\:\mathrm{times}} {\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} …\int_{\mathrm{0}} ^{\:\mathrm{1}} }\:\:{x}_{\mathrm{1}} ^{\alpha} {x}_{\mathrm{2}} ^{\alpha} ….{x}_{{n}} ^{\alpha} \mathrm{ln}\left({x}_{\mathrm{1}} \right)\mathrm{ln}\left({x}_{\mathrm{2}} \right)….\mathrm{ln}\left({x}_{{n}} \right)\mathrm{d}{x}_{\mathrm{1}} \mathrm{d}{x}_{\mathrm{2}} ..\mathrm{d}{x}_{{n}} \\ $$$$\overset{\mathrm{Equal}} {=}\:\:\:\frac{\left(−\right)^{{n}} }{\left(\alpha+\mathrm{1}\right)^{\mathrm{2}{n}} } \\ $$
Commented by mr W last updated on 29/Apr/25
$$\Rightarrow{Q}\mathrm{219552} \\ $$