Question Number 140381 by Willson last updated on 07/May/21
![f(x)=xe^(1−x) Prove that lim_(n→+∞) (√n) ∫^( 1) _0 [f(x)]^n dt = (√(π/2))](Q140381.png)
$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{xe}^{\mathrm{1}−\mathrm{x}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\:\:\:\:\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}}\:\sqrt{\mathrm{n}}\:\underset{\mathrm{0}} {\int}^{\:\mathrm{1}} \left[\mathrm{f}\left(\mathrm{x}\right)\right]^{\mathrm{n}} \:\mathrm{dt}\:=\:\sqrt{\frac{\pi}{\mathrm{2}}} \\ $$