To integrate \( \int \frac{2}{x^2 - 1} \, dx \), we can proceed as follows:

1. **Rewrite the integrand** by factoring the denominator:
   \[
   \int \frac{2}{x^2 - 1} \, dx = \int \frac{2}{(x - 1)(x + 1)} \, dx
   \]

2. **Use partial fraction decomposition**. We can write:
   \[
   \frac{2}{(x - 1)(x + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1}
   \]
   Multiplying both sides by \( (x - 1)(x + 1) \) gives:
   \[
   2 = A(x + 1) + B(x - 1)
   \]
   Expanding and combining like terms, we have:
   \[
   2 = (A + B)x + (A - B)
   \]
   Equating coefficients, we get the system:
   \[
   A + B = 0 \quad \text{and} \quad A - B = 2
   \]
   Solving this system, we find \( A = 1 \) and \( B = -1 \).

3. **Rewrite the integrand using the partial fractions**:
   \[
   \int \frac{2}{(x - 1)(x + 1)} \, dx = \int \left( \frac{1}{x - 1} - \frac{1}{x + 1} \right) \, dx
   \]

4. **Integrate term by term**:
   \[
   = \int \frac{1}{x - 1} \, dx - \int \frac{1}{x + 1} \, dx
   \]
   The integral of \( \frac{1}{x - 1} \) is \( \ln |x - 1| \), and the integral of \( \frac{1}{x + 1} \) is \( \ln |x + 1| \). Thus,
   \[
   = \ln |x - 1| - \ln |x + 1| + C
   \]

5. **Combine the logarithms**:
   \[
   = \ln \left| \frac{x - 1}{x + 1} \right| + C
   \]

### Final Answer:
\[
\int \frac{2}{x^2 - 1} \, dx = \ln \left| \frac{x - 1}{x + 1} \right| + C
\]

Would you like more details on any step or have any questions?

Here are some related questions:
1. How does partial fraction decomposition work in integration?
2. How can logarithmic properties simplify integration results?
3. What if the integrand were \( \frac{2}{x^2 + 1} \) instead?
4. How do we handle integrals with irreducible quadratic denominators?
5. How is the constant of integration determined in indefinite integrals?

**Tip:** Always check if an integrand can be simplified or rewritten before starting integration.

Learn to integrate \( \frac{2}{x^2 - 1} \) with a step-by-step solution involving partial fraction decomposition and logarithmic properties. This problem introduces essential calculus concepts for high school and college students, including the use of partial fractions and properties of logarithmic functions.

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