`int sqrt((cosx-cos^3x)/(1-cos^(3) x)dx` is equal to `Option A: 2/3 sin^(-1)(cos^(3/2)x)+C` `Option B: 3/2 sin^(-1)(cos^(3/2)x)+C` `Option C: 3/2 cos^(-1)(cos^(3/2)x)+C`

Options

A. `(2)/(3)sin^(-1)(cos^(3//2)x)+C`
B. `(3)/(2)sin^(-1)(cos^(3//2)x)+C`
C. `(2)/(3)cos^(-1)(cos^(3//2)x)+C`
D. non of these

Answer

C

Explanation

`intsqrt((cosx-cos^(3)x)/(1-cos^(3)x))dx=int sqrt((cosx)/(1-cos^(3)x))sinxdx`
`=int sqrt((t)/(1-t^(3)))dt`
`=-int (sqrt(t))/(sqrt(1-(t^(3//2))^(2)))dt, " where " t=cosx`
`=-(2)/(3)int((3)/(2)sqrt(t))/(sqrt(1-(t^(3//2))^(2)))dt=(2)/(3)cos^(-1)(t^(3//2))+C`