find-pi-3-pi-2-x-cosx-dx-

Question Number 55996 by maxmathsup by imad last updated on 07/Mar/19

f i n d \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{x}{c o s x} d x

Commented by maxmathsup by imad last updated on 09/Mar/19

l e t I = \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{x}{c o s x} d x c h a n g e m e n t t a n (\frac{x}{2}) = t g i v e

= \int_{\frac{1}{\sqrt{3}}}^{1} \frac{2 a r c t a n (t)}{\frac{1 - t^{2}}{1 + t^{2}}} \frac{2 d t}{1 + t^{2}} = 4 \int_{\frac{1}{\sqrt{3}}}^{1} \frac{a r c t a n t}{1 - t^{2}} d t b u t

\int_{\frac{1}{\sqrt{3}}}^{1} \frac{a r c t a n (t)}{1 - t^{2}} d t = \frac{1}{2} \int_{\frac{1}{\sqrt{3}}}^{1} a r c t a n (t) {\frac{1}{1 - t} + \frac{1}{1 + t}} d t

= \frac{1}{2} \int_{\frac{1}{\sqrt{3}}}^{1} \frac{a r c t a n (t)}{1 - t} d t + \frac{1}{2} \int_{\frac{1}{\sqrt{3}}}^{1} \frac{a r c t a n (t)}{1 + t} d t l e t f (x) = \int_{\frac{1}{\sqrt{3}}}^{1} \frac{a r c t a n (x t)}{1 - t} d t \Rightarrow

f^{^{'}} (x) = \int_{\frac{1}{\sqrt{3}}}^{1} \frac{t}{(1 + x^{2} t^{2}) (1 - t)} d t =_{x t = u} \int_{\frac{x}{\sqrt{3}}}^{x} \frac{u}{x (1 + u^{2}) (1 - \frac{u}{x})} \frac{d u}{x}

= \frac{1}{x} \int_{\frac{x}{\sqrt{3}}}^{x} \frac{u}{(u^{2} + 1) (x - u)} d u l e t d e c o m p o s e F (u) = \frac{u}{(x - u) (u^{2} + 1)}

F (u) = \frac{a}{x - u} + \frac{b u + c}{u^{2} + 1}

a = {l i m}_{u \to x} (x - u) F (u) = \frac{x}{x^{2} + 1}

{l i m}_{u \to + \infty} u F (u) = 0 = - a + b \Rightarrow b = a \Rightarrow F (u) = \frac{a}{x - u} + \frac{a u + c}{u^{2} + 1}

F (0) = 0 = \frac{a}{x} + c \Rightarrow c = - \frac{a}{x} = - \frac{1}{x^{2} + 1} \Rightarrow

F (u) = \frac{x}{(x^{2} + 1) (x - u)} + \frac{\frac{x}{x^{2} + 1} u - \frac{1}{x^{2} + 1}}{u^{2} + 1}

= \frac{x}{x^{2} + 1} {\frac{1}{x - u} + \frac{u - x}{u^{2} + 1}} \Rightarrow \int F (u) d u = \frac{x}{x^{2} + 1} l n ∣ x - u ∣ + \frac{x}{2 (x^{2} + 1)} l n (u^{2} + 1)

- \frac{x^{2}}{x^{2} + 1} a r c t a n (u) + c \Rightarrow

\int_{\frac{x}{\sqrt{3}}}^{x} F (u) d u = \frac{x}{x^{2} + 1} {[l n ∣ x - u ∣ + \frac{1}{2} l n (u^{2} + 1) - x a r c t a n u]}_{\frac{x}{\sqrt{3}}}^{x}

\dots . b e c o n t i n u e d \dots .