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Question Number 58487 by Mr X pcx last updated on 23/Apr/19

l e t f (x) = \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d t}{2 + x s i n t}

1) f i n d a e x p l i c i t f o r m o f f (x)

2) d e t e r m i n e a l s o g (x) = \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{s i n t}{{(2 + x s i n t)}^{2}} d t

3) f i n d t h e v a l u e o f \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d t}{2 + 3 s i n t}

a n d \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{s i n t}{{(2 + 3 s i n t)}^{2}} d t

Commented by maxmathsup by imad last updated on 24/Apr/19

c h a n g e m e n t t a n (\frac{t}{2}) = u g i v e f (x) = \int_{\sqrt{2} - 1}^{\frac{1}{\sqrt{3}}} \frac{2 d u}{(1 + u^{2}) (2 + x \frac{2 u}{1 + u^{2}})}

= \int_{\sqrt{2} - 1}^{\frac{1}{\sqrt{3}}} \frac{2 d u}{2 (1 + u^{2}) + 2 x u} = \int_{\sqrt{2} - 1}^{\frac{1}{\sqrt{3}}} \frac{d u}{u^{2} + 2 x u + 1} = \int_{\sqrt{2} - 1}^{\frac{1}{\sqrt{3}}} \frac{d u}{{(u + x)}^{2} + 1 - x^{2}}

c a s e 1 i f 1 - x^{2} > 0 \Rightarrow∣ x ∣< 1 w e d o t h e c h a n g e m e n t u + x = \sqrt{1 - x^{2}} α

\Rightarrow f (x) = \int_{\frac{\sqrt{2} - 1 + x}{\sqrt{1 - x^{2}}}}^{\frac{\frac{1}{\sqrt{3}} + x}{\sqrt{1 - x^{2}}}} \frac{\sqrt{1 - x^{2}} d α}{(1 - x^{2}) (1 + α^{2})}

= \frac{1}{\sqrt{1 - x^{2}}} {[a r c t a n α]}_{\frac{\sqrt{2} - 1 + x}{\sqrt{1 - x^{2}}}}^{\frac{\frac{1}{\sqrt{3}} + x}{\sqrt{1 - x^{2}}}} \Rightarrow f (x) = \frac{1}{\sqrt{1 - x^{2}}} {a r c t a n (\frac{1 + x \sqrt{3}}{\sqrt{3} \sqrt{1 - x^{2}}}) - a r c t a n (\frac{\sqrt{2} - 1 + x}{\sqrt{1 - x^{2}}})}

c a s e 2 i f 1 - x^{2} < 0 \Rightarrow∣ x ∣> 1 \Rightarrow f (x) = \int_{\sqrt{2} - 1}^{\frac{1}{\sqrt{3}}} \frac{d u}{{(u + x)}^{2} - {(\sqrt{x^{2} - 1})}^{2}}

= \int_{\sqrt{2} - 1}^{\frac{1}{\sqrt{3}}} \frac{d u}{(u + x + \sqrt{x^{2} - 1}) (u + x - \sqrt{x^{2} - 1})}

= \frac{1}{2 \sqrt{x^{2} - 1}} \int_{\sqrt{2} - 1}^{\frac{1}{\sqrt{3}}} {\frac{1}{u + x - \sqrt{x^{2} - 1}} - \frac{1}{u + x + \sqrt{x^{2} - 1}}} d u

= \frac{1}{2 \sqrt{x^{2} - 1}} {[l n ∣ \frac{u + x - \sqrt{x^{2} - 1}}{u + x + \sqrt{x^{2} - 1}} ∣]}_{u = \sqrt{2} - 1}^{u = \frac{1}{\sqrt{3}}}

= \frac{1}{2 \sqrt{x^{2} - 1}} {l n ∣ \frac{\frac{1}{\sqrt{3}} + x - \sqrt{x^{2} - 1}}{\frac{1}{\sqrt{3}} + x + \sqrt{x^{2} - 1}} ∣ - l n ∣ \frac{\sqrt{2} - 1 + x - \sqrt{x^{2} - 1}}{\sqrt{2} - 1 + x + \sqrt{x^{2} - 1}} ∣} .

Commented by maxmathsup by imad last updated on 25/Apr/19

2) w e h a v e f^{^{'}} (x) = \int_{\frac{π}{4}}^{\frac{π}{3}} - \frac{s i n t}{{(2 + x s i n t)}^{2}} d t = - g (x) \Rightarrow g (x) = - f^{^{'}} (x) r e s t t o c a l c u l a t e

f^{^{'}} (x)

Commented by maxmathsup by imad last updated on 25/Apr/19

3) \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d t}{2 + 3 c o s t} = f (3) = \frac{1}{2 \sqrt{3^{2} - 1}} {l n ∣ \frac{\frac{1}{\sqrt{3}} + 3 - \sqrt{3^{2} - 1}}{\frac{1}{\sqrt{3}} + 3 + \sqrt{3^{2} - 1}} ∣ - l n ∣ \frac{\sqrt{2} - 1 + 3 - \sqrt{3^{2} - 1}}{\sqrt{2} - 1 + 3 + \sqrt{3^{2} - 1}} ∣

= \frac{1}{4 \sqrt{2}} {l n ∣ \frac{\frac{1}{\sqrt{3}} + 3 - 2 \sqrt{2}}{\frac{1}{\sqrt{3}} + 3 + 2 \sqrt{2}} ∣ - l n ∣ \frac{2 - \sqrt{2}}{2 + 3 \sqrt{2}} ∣ .