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

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

A

`-1/(6x^3)`

B

`3/(x^2)`

C

`-1/(2x^2)`

D

`-1/(2x^3)`

Text Solution

Verified by Experts

The correct Answer is:
D
`int (1)/(x^(3)(1_x^(6))^(2//3)) dx = xf(x)(1+x^(6))^(1//3) + C`
`= int (1)/(x^7(x^(-6)+1)^(2//3))dx`
Let
`x^(-6)+1=t`
`int (-1/6 dt)/(t^2//3) -6x^(-7)dx = dt`
`=-1/6 int t^(-2//3)dt`
`-1/(2x^2)(1+x^(6))^(1//3)+C`
`:. xf (x) = -1/(2x^2) implies f(x) = -1/(2x^3)`.
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