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The value of the integral overset(1)un...

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

Text Solution

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`int_(1//3)^(1)((x-x^(3))^(1//3))/(x^(4))dx=int_(1//3)^(1)([x^(3)((1)/(x^(2))-1)]^(1//3))/(x^(4))dx=int_(1//3)^(1)(((1)/(x^(2))-1)^(1//3))/(x^(3))dx`
Now, put `(1)/(x^(2)-1=timplies-(2)/(x^(3)dx=dt`. Also , when `x=1`, `t=0` and when `x=1//3`, `t=8`.
So, `int_(1//3)^(1)((x-x^(3))^(1//3))/(x^(4))dx=int_(8)^(0)-(1)/(2)t^(1//3)dt`
`=-(1)/(2)[(3)/(4)t^(4//3)]_(8)^(0)`
`=-(3)/(8)(0-8^(4//3))`
`=6`
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