Question Number 48823 by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18
Commented by Abdulhafeez Abu qatada last updated on 29/Nov/18
$${This}\:{intergrals}\:{i}\:{must}\:{try} \\ $$
Commented by Meritguide1234 last updated on 30/Nov/18
Commented by tanmay.chaudhury50@gmail.com last updated on 30/Nov/18
$${excellent}\:{sir}...\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\int_{{a}} ^{{b}} {f}\left({a}+{b}−{x}\right){dx} \\ $$$$ \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 01/Dec/18
$$\left.\mathrm{7}\right)\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} }<\frac{\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }{dx}<\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{x}^{\mathrm{2}{n}} }{dx} \\ $$$${I}<\mid\frac{{x}^{−\mathrm{2}{n}+\mathrm{1}} }{−\mathrm{2}{n}+\mathrm{1}}\mid_{\mathrm{0}} ^{\mathrm{1}} \\ $$$${I}<\frac{\mathrm{1}}{\mathrm{1}−\mathrm{2}{n}}\left(\mathrm{1}−\mathrm{0}\right) \\ $$$${so}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{{n}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }<\underset{{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{{n}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{dx}}{{x}^{\mathrm{2}{n}} } \\ $$$${I}_{{question}} <\underset{{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{{n}}\:×\frac{\mathrm{1}}{\mathrm{1}−\mathrm{2}{n}} \\ $$$${I}_{{question}} <\underset{{n}\rightarrow\infty} {\mathrm{lim}}\sqrt{{n}}\:×\frac{\frac{\mathrm{1}}{{n}}}{\left(\frac{\mathrm{1}}{{n}}−\mathrm{2}\right)}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\sqrt{{n}}}×\frac{\mathrm{1}}{\left(\frac{\mathrm{1}}{{n}}−\mathrm{2}\right)}=\mathrm{0} \\ $$$${pls}\:{check}... \\ $$$$ \\ $$