∫ x² + 1 / x² − 1 dx is equal to

• x + log | x + 1 | / | x − 1 | + C
• x + log | x − 1 | / | x + 1 | + C
• log | (x − 1)(x + 1) | + C
• log | x² | + C

The integral you provided, ∫ (x² + 1) / (x² − 1) dx, can be simplified and solved using partial fraction decomposition. Let's break down the steps to find the correct answer.

Step 1: Simplifying the Integral

First, we can rewrite the integrand:

• Notice that x² - 1 can be factored as (x - 1)(x + 1).
• Thus, we can express the integrand as a sum of simpler fractions.

Step 2: Partial Fraction Decomposition

We can express:

(x² + 1) / (x² - 1) = A / (x - 1) + B / (x + 1).

By finding the values of A and B, we can integrate each term separately.

Step 3: Integrating Each Term

After performing the decomposition and integrating, we arrive at:

• x + log |x + 1| / |x - 1| + C
• x + log |x - 1| / |x + 1| + C
• log |(x - 1)(x + 1)| + C
• log |x²| + C

Final Result

Thus, the integral can be expressed in various equivalent forms, but the most common simplified result is:

∫ (x² + 1) / (x² - 1) dx = x + log |x + 1| / |x - 1| + C

Remember, C represents the constant of integration, which can vary based on initial conditions or limits of integration.