Let in x denote the logarithm of x with respect to the base e. Let `S sub R` be the set all points where the function In `(x^(2)-1)` is well- defined. Then the number of function `f: S to R` that are differentiable, satisfy
`f'(x)- " In" (x^(2)-1)` for all `x in S ` and f(2)=0, is

Options

A. 0
B. 1
C. 2
D. infinite

Answer

D

Explanation

`ln(x^(2)-1)`
Define for `(x^(2)-1) gt 0`
`S: x in (-oo,-1)cup (1,oo)`
`f'(x)= In(x^(2)-1)`
`int f'(x)dx= intln(x^(2)-1)dx `
`f(x)=ln(x^(2)-1).x-int(2x.x)/(x^(2)-1)dx`
`=xln(x^(2)-1)-int(2x^(2)-2+2)/(x^(2)-1)dx`
`=xln(x^(2)-1)-2x-2x(1)/(2)ln((x-1)/(x+1))+C`
`f(x)=xln(x^(2)-1)-2x-ln((x-1)/(x+1))+C`
`f(2)=2ln 3-4-l((1)/(3))+C=0 implies 2 ln 3-4+C=0`
C=4-3 In3
`f(x)=xln(x^(2)-1)-2x-" In "((x-1)/(x+1))+4-3 ` In 3 defined for S
Infinite C values possibles in set S such that `f'(x)= In(x^(2)-1)`