Question

integral 0-pi [x/(1+ cos(a) sin(x))]

Answer 1

convert x into half angle- like (x/2)
put cos x=cos2x/2-sin2x/2 [because cos2x= cos2x- sin2x]
write (1+cosx) as 2cos2x/2 [ cos 2x+1= 2 cos2 x/2]
and sinx as( 2 sin x/2 cos x/2) [because sin 2x= 2sinx cosx]
then solve it
I =∫(cos2x/2- sin2x/2) /2 cos2x/2 +2 sin x/2 cosx/2 [ limit o to pi/2]
I= 1/2∫(cos x/2+sin x/2) ( cosx/2- sin x/2) / cos x/2( cosx/2 +sin x/2) [limit from 0 tp pi/2]
{ take 1/2 cosx x/2 common from denominator}
I =1/2∫ (cos x/2 -sin x/2) /cos x/2 [limit from 0 to pi/2]
I = 1/2∫ (1- tan x/2)dx [limit from 0 to pi/2]
I =1/2 [x- 2 Log sec x/2] [ limit from 0 to pi/2]
I = 1/2 [pi/2 - log 2]