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Question Number 63667 by mathmax by abdo last updated on 07/Jul/19

1) c a l c u l a t e \int_{0}^{2 π} \frac{d t}{c o s t + x s i n t} w i h x f r o m R .

2) c a l c u l a t e \int_{0}^{2 π} \frac{s i n t}{{(c o s t + x s i n t)}^{2}} d t

3) f i n d [t h e v a l u e o f \int_{0}^{2 π} \frac{d t}{c o s (2 t) + 2 s i n (2 t)}

Commented by mathmax by abdo last updated on 07/Jul/19

1) c h a n g e m e n t z = e^{i t} g i v e

\int_{0}^{2 π} \frac{d t}{c o s t + x s i n t} = \int_{∣ z ∣= 1} \frac{d z}{i z {\frac{z + z^{- 1}}{2} + x \frac{z - z^{- 1}}{2 i}}}

= \int_{∣ z ∣= 1} \frac{d z}{\frac{{i z}^{2} + i}{2} + \frac{{x z}^{2} - x}{2}} = \int_{∣ z ∣= 1} \frac{2 d z}{(x + i) z^{2} - x + i}

= \frac{2}{x + i} \int_{∣ z ∣= 1} \frac{d z}{z^{2} - \frac{x - i}{x + i}} w e h a v e \frac{x - i}{x + i} = \frac{\sqrt{x^{2} + 1} e^{i a r c t a n (- \frac{1}{x})}}{\sqrt{x^{2} + 1} e^{i a r c t a n (\frac{1}{x})}}

= e^{- 2 i a r c t a n (\frac{1}{x})} (w e s u p p o s e x \neq 0) \Rightarrow \sqrt{\frac{x - i}{x + i}} = e^{- i a r c t a n (\frac{1}{x})}

l e t w (z) = \frac{1}{z^{2} - \frac{x - i}{x + i}} \Rightarrow W (z) = \frac{1}{(z - e^{- i a r c t a n (\frac{1}{x})}) (z + e^{- i a r c t a n (\frac{1}{x})})}

r e s i d u s t h e o r e m g i v e

\int_{∣ z ∣= 1} W (z) d z = 2 i π {R e s (W, e^{- i a r c t a n (\frac{1}{x})}) + R e s (W, - e^{- i a r c t a n (\frac{1}{x})})

R e s (W, e^{- i a r c t a n (\frac{1}{x})}) = \frac{1}{2 e^{- i a r c t a n (\frac{1}{x})}} = \frac{1}{2} e^{i a r c t a n (\frac{1}{x})}

R e s (W, - e^{- i a r c t a n (\frac{1}{x})}) = \frac{1}{- 2 e^{- i a r c t a n (\frac{1}{x})}} = - \frac{1}{2} e^{i a r c t a n (\frac{1}{x})} \Rightarrow

t h e r e s i d u s a r e o p p o s i t e s \Rightarrow \int_{∣ z ∣= 1} W (z) d z = 0 \Rightarrow

\forall x \neq 0 \int_{0}^{2 π} \frac{d t}{c o s t + i s i n t} = 0

i f x = 0 w e g e t \int_{0}^{2 π} \frac{d t}{c o s t} = \int_{0}^{π} \frac{d t}{c o s t} + \int_{π}^{2 π} \frac{d t}{c o s t}

\int_{π}^{2 π} \frac{d t}{c o s t} =_{t = π + u} \int_{0}^{π} \frac{d u}{- c o s u} = - \int_{0}^{π} \frac{d u}{c o s u} \Rightarrow

\int_{0}^{2 π} \frac{d t}{c o s t} = 0

Commented by mathmax by abdo last updated on 07/Jul/19

2) l e t f (x) = \int_{0}^{2 π} \frac{d t}{c o s t + x s i n t} \Rightarrow f^{^{'}} (x) = - \int_{0}^{2 π} \frac{s i n t}{{(c o s t + x s i n t)}^{2}} d t = 0

(b e c a u s e f (x) = 0) \Rightarrow \int_{0}^{2 π} \frac{s i n t}{{(c o s t + x s i n t)}^{2}} d t = 0

Commented by mathmax by abdo last updated on 07/Jul/19

3) \int_{0}^{2 π} \frac{d t}{c o s (2 t) + 2 s i n (2 t)} =_{2 t = u} \int_{0}^{4 π} \frac{d u}{2 (c o s u + 2 s i n u)}

= \frac{1}{2} \int_{0}^{2 π} \frac{d u}{c o s u + 2 s i n u} + \int_{2 π}^{4 π} \frac{d u}{c o s u + 2 s i n u} = 0 b e c a u s e

2 π i s p e r i o d .