let-f-x-0-pi-2-ln-1-xsint-1-xsint-dt-1-find-the-value-of-I-0-pi-2-ln-1-xsint-dt-and-J-0-pi-2-ln-1-xsint-dt-2-find-a-simple-form-of-f-x-3-developp-f-at-integr-serie-

Question Number 40624 by math khazana by abdo last updated on 25/Jul/18

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

1) f i n d t h e v a l u e o f I = \int_{0}^{\frac{π}{2}} l n (1 - x s i n t) d t

a n d J = \int_{0}^{\frac{π}{2}} l n (1 + x s i n t) d t

2) f i n d a s i m p l e f o r m o f f (x)

3) d e v e l o p p f a t i n t e g r s e r i e