$\int \frac{1}{x^{2}-1} d x$

Step 1 ：Perform partial fraction decomposition: \(\frac{1}{x^2 - 1} = \frac{1/2}{x - 1} - \frac{1/2}{x + 1}\)

Step 2 ：Integrate each part separately: \(\int \frac{1}{x^{2}-1} d x = \boxed{\frac{1}{2} \ln|x^2 - 1| + C}\)