Question

The value of \[ \int \frac{x^2}{(x^2 + 2)(x^2 + 3)} \, dx \] is equal to:

• A. \( -\sqrt{2} \tan^{-1} \left( \frac{x}{\sqrt{2}} \right) + \sqrt{3} \tan^{-1} \left( \frac{x}{\sqrt{3}} \right) + C \)
• B. \( \sqrt{2} \tan^{-1} \left( \frac{x}{\sqrt{2}} \right) + \sqrt{3} \tan^{-1} \left( \frac{x}{\sqrt{3}} \right) + C \)
• C. \( -\sqrt{2} \tan^{-1} \left( \frac{x}{\sqrt{2}} \right) + \sqrt{3} \tan^{-1} \left( \frac{x}{\sqrt{3}} \right) x + C \)
• D. None of the above

Hint:

For integrals involving rational functions with quadratic denominators, use partial fraction decomposition to simplify the expression before integrating.

Answer 1

The Correct Option is A