We know that

sin² (x/2) + cos² (x/2) = 1

and 2*sin (x/2)*cos (x/2) = sin x

Now, given,

√(1 - sin x) = √{sin² (x/2) + cos² (x/2) - 2*sin (x/2)*cos (x/2)}

                 = √{sin (x/2) + cos (x/2)}²

                 = sin (x/2) + cos (x/2)

Now, ∫ √(1 - sin x) dx = ∫ {sin (x/2) + cos (x/2)} dx

                                  = 2cos (x/2) - 2sin (x/2) + C   where C is an integral constant