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Question Number 55637 by gunawan last updated on 01/Mar/19

$$\mathrm{Value}\mathrm{of}\underset{n\to \mathrm{\infty }}{\lim }n{\int }_0^1\frac{2x^n}{x+x^{2n+1}}dx=..$$

Answered by tanmay.chaudhury50@gmail.com last updated on 01/Mar/19

$${\int }_0^1\frac{2x^{n-1}}{1+{\left(x^n\right)}^2}dx$$$$\frac{2}{n}\times {\int }_0^1\frac{d\left(x^n\right)}{1+{\left(x^n\right)}^2}$$$$\frac{2}{n}\times \mid {tan}^{-1}\left(x^n\right){\mid }_0^1$$$$\frac{2}{n}\times \frac{\pi }{4}$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{\pi }{2n}\times n\to \frac{\pi }{2}$$

Answered by tm888 last updated on 01/Mar/19

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