Question

Let In $x$ denote the logarithm of $x$ with respect to the base $e$. Let $S \subset R $ be the set of all points where the function ln $(x^{2}-1)$ is well- defined . Then, the number of functions $f : S \to R$ that are differentiable, satisfy $f'(x)=$ ln $\left(x^{2}-1\right)$ for all $x \in S$ and $f (2)=0$, is

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (D)

We have,
$f'(x)= \ln (x^{2}+1)$
$f (x) = \int \ln (x^{2}-1)dx$
$ f (x) =x \ln (x^{2}-1) - \int \frac{2x^{2}}{x^{2}-1} dx$
$ f (x) =x \ln (x^{2}-1)-2 \int \left(\frac{x^{2}-1}{x^{2}-1}+\frac{1}{x^{2}-1}\right)dx$
$f \left(x\right)=x \ln \left(x^{2}-1\right)-2x- \ln \left(\frac{x-1}{x+1}\right)+C$
$f (2)=2 \ln (3)-4- \ln \left(\frac{1}{3}\right)+C=0$
$[ \because f (2)=0]$
$\Rightarrow C=4-3\ln 3$
$\therefore f (x)-x \ln (x^{2}-1)-2n \ln \left(\frac{x-1}{x+1}\right)+4-3 \ln 3$
defined for $S$ infinite $C$ values possible in set $S$ such that $f' (x)= \ln (x^{2}-1)$