Question Number 27497 by abdo imad last updated on 07/Jan/18
![let give I_n = n ∫_1 ^(1+(1/n)) f(x^n )dx with f is numerical function integrable on[1,e] .prove that lim_(n−>∝) I_n = ∫_1 ^e ((f(t))/t) dt.](https://www.tinkutara.com/question/Q27497.png)
$${let}\:{give}\:{I}_{{n}} =\:{n}\:\int_{\mathrm{1}} ^{\mathrm{1}+\frac{\mathrm{1}}{{n}}} {f}\left({x}^{{n}} \right){dx}\:{with}\:{f}\:{is}\:{numerical} \\ $$$${function}\:{integrable}\:{on}\left[\mathrm{1},{e}\right]\:.{prove}\:{that} \\ $$$${lim}_{{n}−>\propto} \:\:{I}_{{n}} \:=\:\int_{\mathrm{1}} ^{{e}} \:\:\frac{{f}\left({t}\right)}{{t}}\:{dt}. \\ $$