Step-by-Step Solution
Key Concept: Simplify logarithmic expressions using properties, then apply partial fraction decomposition to reduce the integral to standard forms.
We have $\int \ln\left(\frac{x+1}{x-1}\right)^2 dx = \int \frac{x^2+1}{x^2-1} dx$ after simplification. Using partial fractions: $\frac{x^2+1}{x^2-1} = 1 + \frac{2}{x^2-1} = 1 + \frac{1}{x-1} - \frac{1}{x+1}$. Therefore $\int \left(1 + \frac{1}{x-1} - \frac{1}{x+1}\right) dx = x + \ln|x-1| - \ln|x+1| + C = x + \ln\left|\frac{x-1}{x+1}\right| + C$.
Correct Answer: 1