Question Number 55637 by gunawan last updated on 01/Mar/19
$$\mathrm{Value}\:\mathrm{of}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{n}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{2}{x}^{{n}} }{{x}+{x}^{\mathrm{2}{n}+\mathrm{1}} }\:{dx}=.. \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 01/Mar/19
$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{2}{x}^{{n}−\mathrm{1}} }{\mathrm{1}+\left({x}^{{n}} \right)^{\mathrm{2}} }{dx} \\ $$$$\frac{\mathrm{2}}{{n}}×\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{d}\left({x}^{{n}} \right)}{\mathrm{1}+\left({x}^{{n}} \right)^{\mathrm{2}} } \\ $$$$\frac{\mathrm{2}}{{n}}×\mid{tan}^{−\mathrm{1}} \left({x}^{{n}} \right)\mid_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$\frac{\mathrm{2}}{{n}}×\frac{\pi}{\mathrm{4}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\pi}{\mathrm{2}{n}}×{n}\rightarrow\frac{\pi}{\mathrm{2}} \\ $$
Answered by tm888 last updated on 01/Mar/19
