find-A-n-1-n-1-x-x-arctan-x-1-x-dx-then-calculate-lim-n-A-n-

Question Number 37813 by prof Abdo imad last updated on 17/Jun/18

f i n d A_{n} = \int_{\frac{1}{n}}^{1} x \sqrt{x} a r c t a n (x + \frac{1}{x}) d x

t h e n c a l c u l a t e {l i m}_{n \to + \infty} A_{n} .

Answered by tanmay.chaudhury50@gmail.com last updated on 18/Jun/18

\int {t a n}^{- 1} (x + \frac{1}{x}) \times x^{\frac{3}{2}} d x

{t a n}^{- 1} (x + \frac{1}{x}) \times \frac{x^{\frac{5}{2}}}{\frac{5}{2}} - \int \frac{1}{1 + x^{2} + 2 + \frac{1}{x^{2}}} \times \frac{x^{\frac{5}{2}}}{\frac{5}{2}} d x

d o - \frac{2}{5} \int \frac{x^{\frac{5}{2}}}{x^{2} + \frac{1}{x^{2}} + 3} d x

d 0 - \frac{2}{5} \int \frac{x^{\frac{9}{2}}}{x^{4} + 3 x^{2} + 1} d x

x = t^{2} d x = 2 t d t

d o - \frac{2}{5} \int \frac{t^{2 \times \frac{9}{2}} \times 2 t d t}{t^{8} + 3 t^{4} + 1}

d o - \frac{4}{5} \int \frac{t^{10}}{t^{8} + 3 t^{4} + 1} d t

d o - \frac{4}{5} \int \frac{t^{10} + 3 t^{6} + t^{2} - 3 t^{6} - t^{2}}{t^{8} + 3 t^{4} + 1} d t

d o - \frac{4}{5} \int t^{2} d t + \frac{4}{5} \int \frac{3 t^{6} + t^{2}}{t^{8} + 3 t^{4} + 1} d t

d o - \frac{4}{5} \int t^{2} d t + \frac{4}{5} \int \frac{3 t^{2} + \frac{1}{t^{2}}}{t^{4} + 3 + \frac{1}{t^{4}}}

d o - \frac{4}{5} \int t^{2} d t + \frac{4}{5} \int \frac{3 (t^{2} + \frac{1}{t^{2}}) - \frac{2}{t^{2}}}{{(t^{2} + \frac{1}{t^{2}})}^{2} + 1}

c o n t d