Evaluate: `int(sin^(-1)x)^2\ dx`

Options

A. `(2sin^(-1)x)/(sqrt(1-x^(2)))+C`
B. `(1)/(3)(sin^(-1)x)^(3)+(1)/(sqrt(1-x^(2)))+C`
C. `x(sin^(-1)x)^(2)+(sin^(-1)x)sqrt(1-x^(2))+2x+C`
D. `x(sin^(-1)x)^(2)(sin^(-1)x)sqrt(1-x^(2))-2x+C`

Answer

D

Explanation

Putting x=sin t and dx=cos t dt, we get
`I=int{sin^(-1)(sint)}^(2)costdt=intunderset(I)(t^(2))underset(II)(cost)dt`
`=t^(2)sint-2intunderset(I)underset(II)(sint)dt=t^(2)sint-2[t(-cost)-int(-cost)dt]`
`=t^(2)sint+2tcost-2sint+C=x(sin^(-1)x)^(2)+2(sin^(-1)x)sqrt(1-x^(2))-2x+C`.