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

Step 1: Setting up the partial fractions decomposition.
We are given the integral: \[ I = \int \frac{x^2}{(x^2 + 2)(x^2 + 3)} \, dx \] We can perform partial fraction decomposition to break this down into simpler fractions. The goal is to express: \[ \frac{x^2}{(x^2 + 2)(x^2 + 3)} = \frac{A}{x^2 + 2} + \frac{B}{x^2 + 3} \] Multiplying both sides by \( (x^2 + 2)(x^2 + 3) \), we get: \[ x^2 = A(x^2 + 3) + B(x^2 + 2) \]

Step 2: Solving for constants \( A \) and \( B \).
Expanding both sides: \[ x^2 = A(x^2 + 3) + B(x^2 + 2) \] \[ x^2 = A x^2 + 3A + B x^2 + 2B \] Equating the coefficients of \( x^2 \), we get: \[ A + B = 1 \quad \text{and} \quad 3A + 2B = 0 \] Solving this system of equations: \[ A = -\sqrt{2}, \quad B = \sqrt{3} \]

Step 3: Writing the integral.
Now, we substitute these values of \( A \) and \( B \) into the partial fractions expression: \[ I = \int \frac{-\sqrt{2}}{x^2 + 2} \, dx + \int \frac{\sqrt{3}}{x^2 + 3} \, dx \]

Step 4: Integrating each term.
Each of these integrals is a standard inverse tangent form: \[ \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) \] Thus: \[ I = -\sqrt{2} \tan^{-1} \left( \frac{x}{\sqrt{2}} \right) + \sqrt{3} \tan^{-1} \left( \frac{x}{\sqrt{3}} \right) + C \]

Step 5: Conclusion.
The final solution is: \[ \boxed{-\sqrt{2} \tan^{-1} \left( \frac{x}{\sqrt{2}} \right) + \sqrt{3} \tan^{-1} \left( \frac{x}{\sqrt{3}} \right) + C} \] This matches option (A), which is the correct answer.