Question Number 207547 by SANOGO last updated on 18/May/24
$${lim}\rightarrow+{oo}\:\int_{\mathrm{1}} ^{+{oo}} \:\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}+\mathrm{2}} }{dx}\:=? \\ $$
Commented by SWPlaysMC last updated on 18/May/24
$$\mathrm{Just}\:\mathrm{so}\:\mathrm{you}\:\mathrm{know},\:\mathrm{the}\:\mathrm{infinity}\:\mathrm{symbol}\:\mathrm{is}\:\infty\:\mathrm{and}\:\mathrm{not}\:“{oo}'' \\ $$$$\mathrm{The}\:\mathrm{symbol}\:\mathrm{is}\:\mathrm{in}\:\mathrm{the}\:\int\:\mathrm{menu}\:\left(\mathrm{on}\:\mathrm{the}\:\mathrm{keyboard}\right),\:\mathrm{first}\:\mathrm{page} \\ $$
Answered by Frix last updated on 18/May/24
$$\underset{\mathrm{1}} {\overset{\infty} {\int}}\frac{{x}^{{n}} }{{x}^{{n}+\mathrm{2}} +\mathrm{1}}{dx}\:\overset{{t}=\frac{\mathrm{1}}{{x}}} {=}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{dt}}{{t}^{{n}+\mathrm{2}} +\mathrm{1}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{dt}}{{t}^{{n}+\mathrm{2}} +\mathrm{1}}\:=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{dt}}{{t}^{{n}} +\mathrm{1}}\:=\mathrm{1} \\ $$