find-0-pi-2-ln-1-xsin-2-t-sin-2-t-dt-with-1-lt-x-lt-1-

Question Number 27502 by abdo imad last updated on 07/Jan/18

f i n d \int_{0}^{\frac{π}{2}} \frac{l n (1 + {x s i n}^{2} t)}{{s i n}^{2} t} d t w i t h - 1 < x < 1 .

Commented by abdo imad last updated on 09/Jan/18

l e t p u t f (x) = \int_{0}^{\frac{π}{2}} \frac{l n (1 + {x s i n}^{2} t)}{{s i n}^{2} t} d t a f t e r v e r i f y i n g t h a t f i s

d e r i v a b l e o n] - 1, 1 [w e h a v e f^{,} (x) = \int_{0}^{\frac{π}{2}} \frac{d t}{1 + {x s i n}^{2} t}

b e c a u s e o f / {x s i n}^{2} t / < 1

f^{^{'}} (x) = \int_{0}^{\frac{π}{2}} (\sum_{n = 0}^{\propto} {(- 1)}^{n} x^{n} {s i n}^{2 n} t) d t

= \sum_{n = 0}^{\propto} {(- 1)}^{n} x^{n} \int_{0}^{\frac{π}{2}} {s i n}^{2 n} t d t = \sum_{n = 0}^{\propto} {(- 1)}^{n} W_{n} x^{n}

w i t h W_{n} = \int_{0}^{\frac{π}{2}} {s i n}^{2 n} t d t a n d t h e v a l u e o f W_{n} i s k n o w n

(w a l l i s s i n t e g r a l)

f (x) = \sum_{n = 0}^{\propto} \frac{{(- 1)}^{n}}{n + 1} W_{n} x^{n + 1} + λ

λ = f (0) = 0 \Rightarrow f (x) = \sum_{n = 0}^{\propto} \frac{{(- 1)}^{n}}{n + 1} W_{n} . x^{n + 1} .