The given function is sin 2 x 1+cosx .
∫ sin 2 x 1+cosx dx (1)
Also, sinx=2sin x 2 cos x 2 and 1+cosx=2 cos 2 x 2 (2)
From (1) and (2),
sin 2 x 1+cosx = ( 2sin x 2 cos x 2 ) 2 2 cos 2 x 2 =2 sin 2 x 2 =1−cosx (3)
Put (3) in (1), we get,
∫ sin 2 x 1+cosx dx = ∫ ( 1−cosx )dx = ∫ 1dx− ∫ cosxdx =x−sinx+c
Thus, the integral of the function sin 2 x 1+cosx is x−sinx+c.