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