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let-f-x-x-x-2-dt-lnt-with-x-gt-0-and-x-1-1-prove-that-x-gt-1-x-x-2-xdt-tlnt-f-x-x-x-2-x-2-dt-tlnt-after-find-lim-x-1-f-x-2-calculate-f-x-




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

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