Given $f(x)=\frac{1-x}{e^{x-2}}$, prove that: $$-2<\ln x \cdot \int_{0}^{x} f(t) {\rm{d}}t<\sqrt{x}$$
It is easy for me to show $\int_{0}^{x} f(t) {\rm{d}}t = x {\rm{e}}^{2-x}$, but then I have no idea about what to do next. It seems too tight at some points and not friendly for manual calculation.
Common constants such as the value of the natural constant $e$ can be used for estimation.
