Question Number 32043 by abdo imad last updated on 18/Mar/18
$${let}\:{f}\left({x}\right)=\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\frac{{dt}}{{lnt}}\:\:{with}\:{x}>\mathrm{0}\:{and}\:{x}\neq\mathrm{1} \\ $$ $$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\:{x}>\mathrm{1}\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\frac{{xdt}}{{tlnt}}\:\leqslant{f}\left({x}\right)\leqslant\:\int_{{x}} ^{{x}^{\mathrm{2}} } \:\frac{{x}^{\mathrm{2}} {dt}}{{tlnt}}\:\:{after} \\ $$ $${find}\:{lim}_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right) \\ $$ $$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({x}\right)\:. \\ $$