integration

write cos 2x as 1 - 2sin²x  so we get numenator of above expression as 2sin²x - (1 - cosx)  so the final expression is  sin²x / sin²x/2 - 1 (as 1 - cosx = 2 sin²(x/2))

as sin 2a = 2 sin a cos a  so sin²x  = { 2 sin (x/2) cos (x/2) } ² = 4 sin ²(x/2) cos² (x/2)

so final integration term is  4 cos² (x/2)  - 1  = 2(1 + cos x)  - 1  = 2cos x + 1 

so answer is 2 sinx + x