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Evaluate int(dx)/(x^(2)(x^(4)+1)^((3)/(4...

Evaluate `int(dx)/(x^(2)(x^(4)+1)^((3)/(4)))`.

A

`((x^(4)+1)/(x^(4)))^((1)/(4))+c`

B

`(x^(4)+1)^((1)/(4))+c`

C

`-(x^(4)+1)^((1)/(4))+c`

D

`-((x^(4)+1)/(x^(4)))^((1)/(4))+c`

Text Solution

Verified by Experts

The correct Answer is:
D
`int(dx)/(x^(2)(x^(4)+1)^(3//4))=int(dx)/(x^(5)(1+(1)/(x^(4)))^(3//4))`
Put `1+(1)/(x^(4))=t^(4)`
`rArr" "(-4)/(x^(5))dx = 4t^(3)dt`
`rArr" "(dx)/(x^(5))=-t^(3)dt`
Hence, the integral becomes `int(-t^(3)dt)/(t^(3))=-int dt =-t+c=-(1+(1)/(x^(4)))^(1//4)+c`
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