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(D) The function $f^{\prime}(x) = \ln(x^2-1)$ is defined when $x^2-1 > 0$,which implies $x \in (-\infty, -1) \cup (1, \infty)$.Since $S = (-\infty, -1

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(D) The function $f^{\prime}(x) = \ln(x^2-1)$ is defined when $x^2-1 > 0$,which implies $x \in (-\infty, -1) \cup (1, \infty)$.Since $S = (-\infty, -1) \cup (1, \infty)$ is a disconnected set consisting of two disjoint intervals,the constant of integration can be chosen independently for each interval.Let $f(x) = \int \ln(x^2-1) dx + C_1$ for $x \in (1, \infty)$ and $f(x) = \int \ln(x^2-1) dx + C_2$ for $x \in (-\infty, -1)$.Given $f(2) = 0$,we can determine $C_1$ uniquely.However,there is no condition linking the value of the function on the interval $(1, \infty)$ to the interval $(-\infty, -1)$.Since the constant $C_2$ can be any real number,there are infinitely many such functions $f$.

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