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Question Number 55373 by gunawan last updated on 23/Feb/19

$$\underset{n\to \propto }{\lim }{\int }_0^1\frac{x^n{e}^{x^n}}{\cos x}dx=...$$

Commented by turbo msup by abdo last updated on 23/Feb/19

$$let{I}_n={\int }_0^1\frac{x^n{e}^{x^n}}{cosx}dx\Rightarrow$$$$I_n={\int }_R\frac{x^n{e}^{x^n}}{cosx}{\chi }_{[0,1]}\left(x\right)dx$$$$but{lim}_{n\to +\mathrm{\infty }}\frac{x^n{e}^{x^n}}{cosx}{\chi }_{[0,1]}\left(x\right)=0\Rightarrow$$$${lim}_{n\to +\mathrm{\infty }}{I}_n=0.$$

Answered by tanmay.chaudhury50@gmail.com last updated on 23/Feb/19

$$f\left(x\right)=\frac{x^n{e}^{x^n}}{cosx}$$$$f\left(0\right)=0f\left(1\right)=\frac{e}{cos1}$$$$let{\left[f\left(x\right)\right]}_{max}=Mwhenx[0,1]$$$${\left[f\left(x\right)\right]}_{min}=mwhenx[0,1]$$$$M>f\left(x\right)>m$$$${\int }_0^{1}Mdx>{\int }_0^{1}f\left(x\right)dx>{\int }_0^{1}mdx$$$$M>{\int }_0^{1}f\left(x\right)dx>m$$$$f\left(x\right)=\frac{x^n{e}^{x^n}}{cosx}$$$$1)cosx\ne 0inx[0,1]$$$$2)asn\to \mathrm{\infty }{e}^{x^n}\to 1\left(attachinggraph\right)$$$$3)asn\to \mathrm{\infty }{x}^n\to 0$$$$so\underset{n\to \mathrm{\infty }}{\lim }{\int }_0^1\frac{x^n{e}^{x^n}}{cosx}dx\to 0$$

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