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Question Number 207547 by SANOGO last updated on 18/May/24

$$lim\to +oo{\int }_1^{+oo}\frac{x^n}{1+x^{n+2}}dx=?$$

Commented by SWPlaysMC last updated on 18/May/24

$$\mathrm{Just}\mathrm{so}\mathrm{you}\mathrm{know},\mathrm{the}\mathrm{infinity}\mathrm{symbol}\mathrm{is}\mathrm{\infty }\mathrm{and}\mathrm{not}‘‘{oo}^{″}$$$$\mathrm{The}\mathrm{symbol}\mathrm{is}\mathrm{in}\mathrm{the}\int \mathrm{menu}\left(\mathrm{on}\mathrm{the}\mathrm{keyboard}\right),\mathrm{first}\mathrm{page}$$

Answered by Frix last updated on 18/May/24

$$\underset{1}{\stackrel{\mathrm{\infty }}{\int }}\frac{x^n}{x^{n+2}+1}dx\stackrel{t=\frac{1}{x}}{=}\underset{0}{\stackrel{1}{\int }}\frac{dt}{t^{n+2}+1}$$$$\underset{n\to \mathrm{\infty }}{\lim }\underset{0}{\stackrel{1}{\int }}\frac{dt}{t^{n+2}+1}=\underset{n\to \mathrm{\infty }}{\lim }\underset{0}{\stackrel{1}{\int }}\frac{dt}{t^n+1}=1$$

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