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How could we find the large-$n$ asymptotic of $$\int_{1}^{e^n}\left(1-\frac{\ln x}{n}\right)^n\,dx.$$ I have a suspicion that this is $\sqrt{n}$.

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Denote $t=\ln(x)/n$, then $t$ varies from 0 to 1 and the the integral reads as $$n\int_0^1 ((1-t) e^{t})^ndt.$$ We have $(1-t)e^t=(1-t)(1+t+t^2/2+\ldots)=1-t^2/2+O(t^3)=\exp(-t^2/2+O(t^3))$ for small $t$ and $(1-t)e^t<1$ for $0<t\leqslant 1$. Thus by Laplace method the asymptotics is the same as for the integral $$n\int_0^1e^{-nt^2/2}dt=\sqrt{n}\int_0^{\sqrt{n}}e^{-s^2/2}ds\sim \sqrt{\frac{\pi n}2}.$$

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