Evaluate the following integral: `int_(1//3)^1((x-x^3)^(1//3))/(x^4)dx`

Answer

a

Explanation

`"Let I "=int_(1//3)^(1)((x-x^(3))^(1/3))/(x^(4))dx`
`=int_(1//3)^(1)((x^(3))^(1//3)((x)/(x^(3))-(x^(3))/(x^(3)))^(1//3))/(x^(4))dx`
`=int_(1//3)^(1)(((1)/(x^(2))-1))/(x^(3))dx`
`" Let " (1)/(x^(2))=t rArr (-2)/(x^(3)) dx=dt rArr (dx)/(x^(3)) =(dt)/((-2))`
`x=1 rArr t=1 `
` "and " x=(1)/(3) rArr t=3^(2)=9`
`:. I= int_(9)^(1) (t-1)^(1//3) (dt)/((-2)) =(1)/(2) [((t-1)^(1/3+1))/((1)/(3)+1)]_(9)^(1)`
`= -(1)/(2) xx(3)/(4) [(t-1)^(4/3)]_(9)^(1) =-(3)/(8)[(1-1)^(4/3)-(9-1)^(4/3)]`
`=-(3)/(8)[0-(2^(3))^(4/3)]=-(3)/(8)xx(-16)=6`