`intsqrt((x)/(1-x^(3)))dx` is equal to

Options

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

Answer

A

Explanation

`l=intsqrt((x)/(1-x^(3)))dx=int(sqrtx)/(sqrt(1-x^(3)))dx=int(sqrtx)/(sqrt(1-(x^(3//2))^(2)))dx`
`"Put "x^(3//2)=t rArr sqrtx dx =(2)/(3)dt`
`"So, "l=(2)/(3)int(dt)/(sqrt(1-t^(2)))=(2)/(3)sin^(-1)(x^(3//2))+C`