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"If"int(dx)/(x^(3)(1+x^(6))^(2/3))=xf(x)...

`"If"int(dx)/(x^(3)(1+x^(6))^(2/3))=xf(x)(1+x^(6))^(1/3)+C` where, C is a constant of integration, then the function f(x) is equal to

A

`-(1)/(2)`

B

`-(1)/(6)`

C

`-(6)/(x)`

D

`-(x)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
a
Let `=int(1)/(x^(3)(1+x^(6))^(2//3))dx` . Then
`I=int(1)/(7(1+(1)/(x^(6)))^(2//3))dx=-(1)/(6)int(1+(1)/(x^(6)))^(-2//3)d(1+(1)/(x^(6)))`
`rArrI=-(1)/(2)(1+(1)/(x^(6)))^(1//3)+C`
`rArrf(x)(1+x^(-6))^(1//3)+C=-(1)/(2)(1+x^(-6))^(1//3)+C`
`rArrf(x)=-(1)/(2)`.
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