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Problem:

You are asked to compute the definite integral:

$\int_{0}^{\infty} \left( \frac{x^2}{(x^2 + 4)(x^2 + 9)} \right) \, dx$

Solution:

This type of integral is commonly solved using the method of partial fractions.

1. Decompose the integrand into partial fractions:

   The integrand can be decomposed as:

   $\frac{x^2}{(x^2 + 4)(x^2 + 9)} = \frac{A}{x^2 + 4} + \frac{B}{x^2 + 9}$

   To find $A$ and $B$, multiply both sides by $(x^2 + 4)(x^2 + 9)$ and equate coefficients:

   $x^2 = A(x^2 + 9) + B(x^2 + 4)$

   Expanding and combining like terms:

   $x^2 = (A + B)x^2 + (9A + 4B)$

   Comparing coefficients of $x^2$ and the constant term on both sides:

   • Coefficient of $x^2$: $1 = A + B$
   • Constant term: $0 = 9A + 4B$

2. Solve the system of equations:

   From $1 = A + B$ and $0 = 9A + 4B$, solve for $A$ and $B$:

   • From $1 = A + B$, we have $B = 1 - A$.

   • Substitute $B = 1 - A$ into $0 = 9A + 4B$:

     $0 = 9A + 4(1 - A) \Rightarrow 0 = 9A + 4 - 4A \Rightarrow 5A = -4 \Rightarrow A = -\frac{4}{5}$

   • Substituting $A = -\frac{4}{5}$ into $B = 1 - A$:

     $B = 1 + \frac{4}{5} = \frac{9}{5}$

   So the decomposition is:

   $\frac{x^2}{(x^2 + 4)(x^2 + 9)} = \frac{-\frac{4}{5}}{x^2 + 4} + \frac{\frac{9}{5}}{x^2 + 9}$

3. Integrate term by term:

   Now, integrate each term separately:

   $\int_{0}^{\infty} \left( -\frac{4/5}{x^2 + 4} + \frac{9/5}{x^2 + 9} \right) dx$

   These integrals are standard:

   $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C$

   So,

   $\int_{0}^{\infty} \frac{-\frac{4}{5}}{x^2 + 4} dx = -\frac{4}{5} \cdot \frac{1}{2} \left[ \frac{\pi}{2} \right] = -\frac{2\pi}{5}$

   and

   $\int_{0}^{\infty} \frac{\frac{9}{5}}{x^2 + 9} dx = \frac{9}{5} \cdot \frac{1}{3} \left[ \frac{\pi}{2} \right] = \frac{3\pi}{5}$

4. Final Answer:

   The value of the integral is:

   $\frac{3\pi}{5} - \frac{2\pi}{5} = \frac{\pi}{5}$

So, the value of the integral is:

$\boxed{\frac{\pi}{5}}$

Would you like further details or explanations on any part of this solution?

Follow-Up Questions:

1. How is partial fraction decomposition used to simplify integrals?
2. What is the importance of recognizing standard integral forms?
3. Can this integral be solved using any other methods, such as contour integration?
4. How would the integral change if the limits were different, say from $0$ to $1$?
5. What are some practical applications of integrals like this in physics or engineering?

Tip:

When solving definite integrals involving rational functions, always check if the integrand can be simplified using partial fraction decomposition. This technique often turns a complex integral into a sum of simpler ones.