Step-by-step explanation:
[tex]\small{ \displaystyle\tt \int_{0}^{1} \int_{0} ^{ x_{1}} \int_{ 0 }^{x_{2}}... \int_{ 0 }^{x_{99}} ln(1 -x_{100} ) dx_{100}... dx_{1}}[/tex]
[tex]\small{{ \displaystyle\tt = - \int_{0}^{1} \int_{0} ^{ x_{1}} \int_{ 0 }^{x_{2}}... \int_{ 0 }^{x_{98}} \sum_{n = 1}^{ \infty \frac{}{} \frac{}{} } \frac{1}{n} \int_{0}^{x_{99}} {x}^{n}_{100} \: dx_{100}...dx_{1} }}[/tex]
[tex]{{ 0 }^{x_{98}} \frac{{x}^{n + 1}_{99}}{n + 1} \: dx_{99}...dx_{1} }[/tex]
[tex]\small{ \displaystyle\tt = -\sum_{n = 1}^{ \infty \frac{}{} \frac{}{} } \frac{1}{n}\int_{0}^{1} \int_{0} ^{ x_{1}} }[/tex]
[tex]\displaystyle \tt= - \frac{1}{(100)!_{} } \sum_{n = 1}^{ \infty } \frac{(n - 1)!(100)!}{(n + 101)!}=−(100)!1n=1∑∞(n+101)!(n−1)!(100)![/tex]
[tex]\displaystyle \tt= - \frac{1}{(100)!_{} } \sum_{n = 1}^{ \infty} \int_{0}^{1} {x}^{n - 1} (1 - x {)}^{100} [/tex]
[tex]\displaystyle \sf - \dfrac{1}{(100!)} \int_{0}^{1} \dfrac{(1 - x)^{100} }{1 - x} \:[/tex]
[tex]\displaystyle\tt { = - \frac{1}{(100)!} \int_{0}^{1} {x}^{99} \: dx=−(100)!1∫01x99dx}
[/tex]
