Question

Evaluate the following integrals: \[ \int \frac{(x^4 + 1)}{x(2x + 1)^2} \, dx \] and \[ \int \frac{1}{x^4 + 5x^2 + 6} \, dx \]

• A. \( \frac{1}{(2x + 1)} \)
• B. \( \frac{1}{(x^4 + 5x^2 + 6)} \)
• C. \( \frac{1}{2} \left( \ln \left| \frac{x^2 + 3}{x + 2} \right| \right) \)
• D. \( \frac{1}{3} \left( \ln |x^2 + 5x + 6| \right) \)

Hint:

When dealing with integrals of rational functions, look for opportunities to simplify and split into manageable parts, and use substitution when applicable.