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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)/(6x^(3))`

B

`-(1)/(2x^(3))`

C

`-(1)/(2x^(2))`

D

`(3)/(x^(2))`

Text Solution

Verified by Experts

The correct Answer is:
B
Let `I = int(dx)/(x^(3)(1+x^(6))^(2//3))`
`=int(dx)/(x^(3)x^(4)((1)/(x^(6))+1)^(2//3))=int(dx)/(x^(7)((1)/(x^(6))+1)^(2//3))`
Now, put `(1)/(x^(6))+1 = t^(3)`
`rArr" " -(6)/(x^(7))dx = 3t^(2)dt`
`rArr" "(dx)/(x^(7))=-(t^(2))/(2)dt`
So, `I=int(-(1)/(2)t^(2)dt)/(t^(2))= -(1)/(2)int dt`
`= -(1)/(2)t+C = -(1)/(2)((1)/(x^(6))+1)^(1//3)+C" "[therefore t^(3)=(1)/(x^(6))+1]`
`= -(1)/(2)(1)/(x^(2))(1+x^(6))^(1//3)+C`
`= x*f(x)*(1+x^(6))^(1//3)+C" "["given"]`
On comparing both sides, we get
`f(x) = -(1)/(2x^(3))`
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