integration

multiply numinator and denominator with undre root sin x  and write sin x as  1 - cos ² x   so the final integration is  I = sin x /  (sin x)  =  sin x  dx /  ( 1 - cos ² x  )    

now let cos x = t  ===>  - sinx dx = dt ===> I = - dt /  (1 - t ²)   ====>  I = sin ^-1 t =  - sin ^-1 (cos x )