Integrate 1/(x^2 - 1) dx using partial fractions

Problem: Evaluate the indefinite integral $\int \frac{1}{x^2 - 1} \, dx$ using partial fractions.

Steps

1. Factor the denominator
   $$\int \frac{1}{(x - 1)(x + 1)} \, dx$$
2. Apply partial fraction decomposition
   $$\int \left( \frac{1/2}{x - 1} - \frac{1/2}{x + 1} \right) dx$$
3. Integrate each term
   $$\frac{1}{2} \ln|x - 1| - \frac{1}{2} \ln|x + 1| + C$$
4. Combine logarithms using logarithm properties
   $$\frac{1}{2} \ln\left| \frac{x - 1}{x + 1} \right| + C$$