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Evaluate: int(dx)/(x^2(x^6+1)^(5/6))...

Evaluate: `int(dx)/(x^2(x^6+1)^(5/6))`

Text Solution

Verified by Experts

`I=int(dx)/(x^(2)(x^(4)+1)^((3)/(4)))=int(dx)/(x^(2).x^(3)(1+(1)/(x^(4)))^((3)/(4)))`
`=int((1+(1)/(x^(4)))^(-(3)/(4))dx)/(x^(5))`
Let `1+(1)/(x^(4))=t " or " (-4)/(x^(5))dx=dt`
` :. I= -(1)/(4)int t^(-(3)/(4))dt= -(1)/(4)(t^((1)/(4)))/((1)/(4))+C`
`= -(1+(1)/(x^(4)))^((1)/(4))+C`
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