integration

cosx - cos2x = 2cos(3x/2)cos(x/2)        and 1 - cosx = 2cos²(x/2)                                                                                    putting in above integration we get  cos (3x/2)  /  cos(x/2)

we know that cos 3x = 4cos³x - 3cosx   so   cos (3x/2) = 4cos³ (x/2) - 3cos(x/2)                                                                so final integration is  4cos²(x/2) - 3  ===> 2(1 + cosx) - 3   ===>2cosx - 1     which is  2sin x - x