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Question Number 52197 by maxmathsup by imad last updated on 04/Jan/19

c a l c u l a t e \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{c o s x - s i n x}{2 + s i n (2 x)} d x

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Jan/19

2 + s i n 2 x

1 + {s i n}^{2} x + {c o s}^{2} x + 2 s i n x c o s x

1 + {(s i n x + c o s x)}^{2}

\int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d (s i n x + c o s x)}{1 + {(s i n x + c o s x)}^{2}}

∣ {t a n}^{- 1} (s i n x + c o s x) ∣_{\frac{π}{4}}^{\frac{π}{3}}

{t a n}^{- 1} (\frac{\sqrt{3}}{2} + \frac{1}{2}) - {t a n}^{- 1} (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}})

{t a n}^{- 1} (\frac{1 + \sqrt{3}}{2}) - {t a n}^{- 1} (\sqrt{2})

{t a n}^{- 1} (\frac{\frac{1 + \sqrt{3}}{2} - \sqrt{2}}{1 + \frac{1 + \sqrt{3}}{\sqrt{2}}})

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

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

{t a n}^{- 1} (\frac{1 + \sqrt{3} - 2 \sqrt{2}}{2 + \sqrt{2} + \sqrt{6}})

Commented by Abdo msup. last updated on 05/Jan/19

t h a n k y o u s i r T a n m a y .