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The value of the integral `int_(1/3)^(1)((x-x^(3))^(1/3))/(x^(4))dx` is

Answer

2

Explanation

`I=int_(0)^(1//2)(1+sqrt(3))/(((x+1)^(2)(1-x)^(6))^(1//4))dx`
`=int_(0)^(1//2) ((1+sqrt(3))dx)/((1+x)^(2)[((1-x)^(6))/((1+x)^(6))]^(1//4)`
Put `(1-x)/(1+x)=t`
`implies(-2dx)/((1+x)^(2))=dt`
`:.I=int_(1)^(1//3)((1+sqrt(3))dt)/(-2t^(3//2))=(-(1+sqrt(3)))/2xx|(-2)/(sqrt(t))|_(1)^(1//3)`
`=(1+sqrt(3))(sqrt(3)-1)=2`