Question Number 195223 by Rodier97 last updated on 27/Jul/23
![In=∫_1 ^e (((lnx)^n )/x^2 ) dx using an enclosing lnx on interval [1;e] show that ∀n ∈ N^∗ , 0 ≤In≤ 1](https://www.tinkutara.com/question/Q195223.png)
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{In}=\int_{\mathrm{1}} ^{\mathrm{e}} \frac{\left({lnx}\right)^{{n}} }{{x}^{\mathrm{2}} }\:{dx} \\ $$$$\:\:\:\:\:\mathrm{using}\:\mathrm{an}\:\mathrm{enclosing}\:{lnx}\:\mathrm{on}\:\mathrm{interval}\:\left[\mathrm{1};\mathrm{e}\right]\:\mathrm{show}\:\mathrm{that}\:\forall{n}\:\in\:\mathbb{N}^{\ast} ,\:\mathrm{0}\:\leq\mathrm{In}\leq\:\mathrm{1} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$