Question

Evaluate the integral \[ \int_0^\infty \frac{dx}{(x^2 + 4)(x^2 + 9)} = \]

• A. \( \frac{\pi}{120} \)
• B. \( \frac{\pi}{60} \)
• C. \( \frac{\pi}{80} \)
• D. \( \frac{-\pi}{60} \)

Hint:

For integrals involving rational functions, use partial fraction decomposition to simplify the integrand before integrating.

Answer 1

The Correct Option is B

Step 1: Use partial fractions.
We are given the integral \[ \int_0^\infty \frac{dx}{(x^2 + 4)(x^2 + 9)}. \] We can decompose the integrand using partial fractions: \[ \frac{1}{(x^2 + 4)(x^2 + 9)} = \frac{A}{x^2 + 4} + \frac{B}{x^2 + 9}. \]
Step 2: Solve for constants.
Multiplying both sides by \( (x^2 + 4)(x^2 + 9) \), we get: \[ 1 = A(x^2 + 9) + B(x^2 + 4). \] Solving for \( A \) and \( B \), we find the constants. After simplifying, we can integrate the resulting terms.
Step 3: Integration.
Using the standard integrals for terms of the form \( \frac{1}{x^2 + a^2} \), the final result of the integral is \( \frac{\pi}{60} \). Thus, the correct answer is option (B).