Evaluate: `intsqrt(x/(a^3-x^3))dx`

Answer

`(2)/(3)"sin"^(-1)((x^(3//2))/(a^(3//2)))+C`

Explanation

`I=intsqrt((x)/(a^(3)-x^(3)))dx=int(sqrt(x))/(sqrt((a^(3//2))^(2)-(x^(3//2))^(2)))dx`
` "Let " x^(3//2)=t " or "(3)/(2)x^(1//2)dx=dt " or "dx=(2)/(3sqrt(x))dt`
` :. I=int(2//3dt)/(sqrt((a^(3//2))^(2)-t^(2)))dt=(2)/(3)int(dt)/(sqrt((a^(3//2))^(2)-t^(2)))`
`=(2)/(3)"sin"^(-1)((1)/(a^(3//2)))+C=(2)/(3)"sin"^(-1)((x^(3//2))/(a^(3//2)))+C`