Given an integer $n\geq2$, consider the following integral


$$I\_n:=\int\_0^1nx^{n-1}\sqrt{\left\vert \frac{\log(1-x)}{\log n}\right\vert} \, dx.$$


**QUESTION.** Is this true? It appears to be so.
$$\lim\_{n\rightarrow\infty}I\_n=1.$$



Asymptotics of an integral requested
Change variables to $y=n(1-x)$, which gives
$$
I\_n=\int\limits\_{0}^{n}\left(
1-\frac{y}{n}\right)^{n-1}
\sqrt{1-\frac{\log y}{\log n}}\ {\rm d}y\ ,
$$
then dominated convergence should allow passing to the limit inside the integral
so that
$$
\lim\_{n\rightarrow \infty} I\_n=\int\limits\_{0}^{\infty} e^{-y}\ {\rm d}y=1\ .
$$


