Question

Evaluate \[ \int \frac{x^2 + 4}{x^4 + 16} \, dx \]

• A. \( \frac{1}{2\sqrt{2}} \tan^{-1} \left( \frac{x^2 - 4}{2\sqrt{2}} \right) + c \)
• B. \( \frac{1}{2\sqrt{2}} \tan^{-1} \left( \frac{x^2 - 4}{2\sqrt{2}} \right) + c \)
• C. \( \frac{1}{2\sqrt{2}} \tan^{-1} \left( \frac{x^2 - 4}{x^4} \right) + c \)
• D. None of the above

Hint:

Use appropriate substitutions to simplify integrals and find their solutions.

Answer 1

The Correct Option is A

Step 1: Solve the integral.
Use the substitution \( u = x^2 + 4 \) to solve the integral.
Step 2: Conclusion.
Thus, the solution to the integral is \( \frac{1}{2\sqrt{2}} \tan^{-1} \left( \frac{x^2 - 4}{2\sqrt{2}} \right) + c \).
Final Answer: \[ \boxed{A} \]