Question Number 55373 by gunawan last updated on 23/Feb/19
$$\underset{{n}\rightarrow\propto} {\mathrm{lim}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} {e}^{{x}^{{n}} } }{\mathrm{cos}\:{x}}\:{dx}=… \\ $$
Commented by turbo msup by abdo last updated on 23/Feb/19
![let I_n =∫_0 ^1 ((x^n e^x^n )/(cosx))dx ⇒ I_n =∫_R ((x^n e^x^n )/(cosx)) χ_([0,1]) (x)dx but lim_(n→+∞) ((x^n e^x^n )/(cosx)) χ_([0,1]) (x)=0 ⇒ lim_(n→+∞) I_n =0 .](https://www.tinkutara.com/question/Q55398.png)
$${let}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{{n}} {e}^{{x}^{{n}} } }{{cosx}}{dx}\:\Rightarrow \\ $$$${I}_{{n}} =\int_{{R}} \:\:\frac{{x}^{{n}} \:{e}^{{x}^{{n}} } }{{cosx}}\:\chi_{\left[\mathrm{0},\mathrm{1}\right]} \left({x}\right){dx} \\ $$$${but}\:{lim}_{{n}\rightarrow+\infty} \:\:\frac{{x}^{{n}} \:{e}^{{x}^{{n}} } }{{cosx}}\:\chi_{\left[\mathrm{0},\mathrm{1}\right]} \left({x}\right)=\mathrm{0}\:\Rightarrow \\ $$$${lim}_{{n}\rightarrow+\infty} \:{I}_{{n}} =\mathrm{0}\:. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 23/Feb/19
![f(x)=((x^n e^x^n )/(cosx)) f(0)=0 f(1)=(e/(cos1)) let [f(x)]_(max) =M when x[0,1] [f(x)]_(min) =m when x[0,1] M>f(x)>m ∫_0 ^1 Mdx>∫_0 ^1 f(x)dx>∫_0 ^1 mdx M>∫_0 ^1 f(x)dx>m f(x)=((x^n e^x^n )/(cosx)) 1)cosx≠0 in x [0,1] 2)as n→∞ e^x^n →1(attaching graph) 3)as n→∞ x^n →0 so lim_(n→∞) ∫_0 ^1 ((x^n e^x^n )/(cosx))dx→0](https://www.tinkutara.com/question/Q55403.png)
$${f}\left({x}\right)=\frac{{x}^{{n}} {e}^{{x}^{{n}} } }{{cosx}} \\ $$$${f}\left(\mathrm{0}\right)=\mathrm{0}\:\:\:\:{f}\left(\mathrm{1}\right)=\frac{{e}}{{cos}\mathrm{1}} \\ $$$${let}\:\left[{f}\left({x}\right)\right]_{{max}} ={M}\:\:{when}\:\:\:{x}\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left[{f}\left({x}\right)\right]_{{min}} ={m}\:\:\:{when}\:{x}\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\:\:\:\:\:\:{M}>{f}\left({x}\right)>{m} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {Mdx}>\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}>\int_{\mathrm{0}} ^{\mathrm{1}} {mdx} \\ $$$${M}>\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}>{m} \\ $$$${f}\left({x}\right)=\frac{{x}^{{n}} {e}^{{x}^{{n}} } }{{cosx}} \\ $$$$\left.\mathrm{1}\right){cosx}\neq\mathrm{0}\:{in}\:{x}\:\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.\mathrm{2}\right){as}\:{n}\rightarrow\infty\:{e}^{{x}^{{n}} } \rightarrow\mathrm{1}\left({attaching}\:{graph}\right) \\ $$$$\left.\mathrm{3}\right){as}\:{n}\rightarrow\infty\:\:{x}^{{n}} \rightarrow\mathrm{0} \\ $$$${so}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} {e}^{{x}^{{n}} } }{{cosx}}{dx}\rightarrow\mathrm{0} \\ $$