समाकलन `int_((1)/(3))^(1)((x-x^(3))^((1)/(3)))/(x^(4))dx` का मान है :

Explanation

माना `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))^(1//3))/(x^(3))dx`
माना `(1)/(x^(2))=t rArr (-2)/(x^(3))dx=dt" "rArr" "(dx)/(x^(3))=(dt)/((-2))` ltbr `x=1 rArr t=1`
और `x=(1)/(3)rArr t=3^(2)=9`
`therefore" "I=int_(9)^(1)(t-1)^(1//3)(dt)/((-2))=-(1)/(2)[((t-1)^((1)/(3)+1))/((1)/(3)+1)]_(0)^(1)`
`=-(1)/(2)xx(3)/(4)[(t-1)^((4)/(3))]=-(3)/(8)[(1-1)^((4)/(3))-(9-1)^((4)/(3))]`
`=-(3)/(8)[0-(2^(3))^((4)/(3))]=-(3)/(8)xx(-16)=6`