calculate-I-pi-3-pi-2-cos-2x-sin-x-cosx-dx-and-J-pi-3-pi-2-sin-2x-sin-x-cos-x-dx-

Question Number 42628 by prof Abdo imad last updated on 29/Aug/18

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

J = \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{s i n (2 x)}{s i n (x) + c o s (x)} d x

Commented by maxmathsup by imad last updated on 30/Aug/18

w e h a v e I = \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{c o s (2 x)}{\sqrt{2} c o s (x - \frac{π}{4})} d x c h a n g e m e n t x - \frac{π}{4} = t g i v e

I = \int_{\frac{π}{12}}^{\frac{π}{2}} \frac{c o s (2 (t + \frac{π}{4}))}{\sqrt{2} c o s t} d t = \int_{\frac{π}{12}}^{\frac{π}{2}} \frac{- s i n (2 t)}{\sqrt{2} c o s (t)} d t = - \frac{1}{\sqrt{2}} \int_{\frac{π}{12}}^{\frac{π}{2}} \frac{2 s i n t c o s t}{c o s t} d t

= - \sqrt{2} \int_{\frac{π}{12}}^{\frac{π}{2}} s i n (t) d t = - \sqrt{2} {[- c o s t]}_{\frac{π}{12}}^{\frac{π}{2}} = \sqrt{2} {0 - c o s (\frac{π}{12})}

w e h a v e {c o s}^{2} (\frac{π}{12}) = \frac{1 + c o s (\frac{π}{6})}{2} = \frac{1 + \frac{\sqrt{3}}{2}}{2} = \frac{2 + \sqrt{3}}{4} \Rightarrow c o s (\frac{π}{12}) = \frac{\sqrt{2 + \sqrt{3}}}{2} \Rightarrow

I = - \sqrt{2} \sqrt{2 + \sqrt{3}} = - \sqrt{4 + 2 \sqrt{3}} .

Commented by maxmathsup by imad last updated on 30/Aug/18

w e h a v e J = \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{s i n (2 x)}{\sqrt{2} s i n (x + \frac{π}{4})} d x =_{x + \frac{π}{4} = t} \int_{\frac{7 π}{12}}^{\frac{3 π}{4}} \frac{s i n (2 (t - \frac{π}{4}))}{\sqrt{2} s i n t} d t

= \int_{\frac{7 π}{12}}^{\frac{3 π}{4}} \frac{- c o s (2 t)}{\sqrt{2} s i n t} d t = - \int_{\frac{7 π}{12}}^{\frac{3 π}{4}} \frac{1 - 2 {s i n}^{2} t}{\sqrt{2} s i n t} d t = \int_{\frac{7 π}{12}}^{\frac{3 π}{4}} \sqrt{2} s i n t d t - \frac{1}{\sqrt{2}} \int_{\frac{7 π}{12}}^{\frac{3 π}{4}} \frac{d t}{s i n t}

b u t \int_{\frac{7 π}{12}}^{\frac{3 π}{4}} \sqrt{2} s i n t d t = \sqrt{2} {[- c o s t]}_{\frac{7 π}{12}}^{\frac{3 π}{4}} = \sqrt{2} {c o s (\frac{7 π}{12}) - c o s (\frac{3 π}{4})}

= \sqrt{2} {- s i n (\frac{π}{12}) + c o s (\frac{π}{4})} = \sqrt{2} {\frac{1}{\sqrt{2}} - \frac{\sqrt{2 - \sqrt{3}}}{2}} a l s o

\int_{\frac{7 π}{12}}^{\frac{3 π}{4}} \frac{d t}{s i n t} =_{t a n (\frac{t}{2}) = u} \int_{t a n (\frac{7 π}{24})}^{t a n (\frac{3 π}{8})} \frac{1}{\frac{2 u}{1 + u^{2}}} \frac{2 d u}{1 + u^{2}} = {[l n ∣ u ∣]}_{t a n (\frac{7 π}{24})}^{t a n (\frac{3 π}{8})}

= l n ∣ t a n (\frac{3 π}{8}) ∣ - l n ∣ t a n (\frac{7 π}{24}) ∣ \Rightarrow

J = \sqrt{2} {\frac{1}{\sqrt{2}} - \frac{\sqrt{2 - \sqrt{3}}}{2}} - \frac{1}{\sqrt{2}} {l n ∣ t a n (\frac{3 π}{8}) ∣ - l n ∣ t a n (\frac{7 π}{24}) ∣} .