How to Solve the Integral of (x^2024 - 1) / (x^506 - 1)?

Solving the Integral: ∫ (x^2024 - 1) / (x^506 - 1) dx

The integral we are solving is:

∫ (x^2024 - 1) / (x^506 - 1) dx

We start by simplifying the expression using polynomial division. Dividing x²⁰²⁴ - 1 by x⁵⁰⁶ - 1 gives:

                x2024 - 1 = (x506 - 1) * (x1518 + x1012 + x506 + 1)
            

This allows us to rewrite the integral as:

∫ (x¹⁵¹⁸ + x¹⁰¹² + x⁵⁰⁶ + 1) dx

Now, we can integrate each term:

• ∫ x¹⁵¹⁸ dx = x¹⁵¹⁹ / 1519
• ∫ x¹⁰¹² dx = x¹⁰¹³ / 1013
• ∫ x⁵⁰⁶ dx = x⁵⁰⁷ / 507
• ∫ 1 dx = x

Thus, the final result is:

                x1519 / 1519 + x1013 / 1013 + x507 / 507 + x + C
            

Where C is the constant of integration.