1-calculate-F-x-1-x-arctan-t-t-2-dt-with-x-1-2-calculate-A-n-1-n-arctan-t-t-2-dt-and-find-lim-n-A-n-

Question Number 39369 by maxmathsup by imad last updated on 05/Jul/18

1) c a l c u l a t e F (x) = \int_{1}^{\sqrt{x}} \frac{a r c t a n (t)}{t^{2}} d t w i t h x ⩾ 1

2) c a l c u l a t e A_{n} = \int_{1}^{\sqrt{n}} \frac{a r c t a n (t)}{t^{2}} d t a n d f i n d {l i m}_{n \to + \infty} A_{n}

Commented by math khazana by abdo last updated on 07/Jul/18

1) b y p a r t s F (x) = {[- \frac{1}{t} a r c t a n t]}_{1}^{\sqrt{x}} - \int_{1}^{\sqrt{x}} \frac{- 1}{t (1 + t^{2})} d t

= \frac{π}{4} - \frac{a r c t a n (\sqrt{x})}{\sqrt{x}} + \int_{1}^{\sqrt{x}} (\frac{1}{t} - \frac{t}{1 + t^{2}}) d t

= \frac{π}{4} - \frac{a r c t a n (\sqrt{x})}{\sqrt{x}} + l n (\sqrt{x}) - \frac{1}{2} {[l n (1 + t^{2})]}_{1}^{\sqrt{x}}

= \frac{π}{4} - \frac{a r c t a n (\sqrt{x})}{\sqrt{x}} + l n (\sqrt{x}) - \frac{1}{2} {l n (1 + x) - l n (2)}

2) w e h a v e A_{n} = F (n)

= \frac{π}{4} - \frac{a r c t a n (\sqrt{n})}{\sqrt{n}} + l n (\sqrt{n}) - l n \sqrt{1 + n} + \frac{l n (2)}{2}

= \frac{π}{4} + \frac{l n (2)}{2} + l n (\frac{\sqrt{n}}{\sqrt{n + 1}}) - \frac{a r c t a n (\sqrt{n)}}{\sqrt{n}} \Rightarrow

{l i m}_{n \to + \infty} A_{n} = \frac{π}{4} + \frac{l n (2)}{2} .