The problem asks you to evaluate the improper integral

$\int_{0}^{\infty} \frac{2 \, dx}{9 + x^2}$

or determine if it is divergent.

Solution:

The integral has the form:

$\int_{0}^{\infty} \frac{2 \, dx}{a^2 + x^2}$

This is a standard integral with the result:

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

In our case, $a = 3$ since $9 = 3^2$, and the coefficient outside the integral is 2. Applying the formula:

$\int_{0}^{\infty} \frac{2 \, dx}{9 + x^2} = \frac{2}{3} \left[ \tan^{-1} \left( \frac{x}{3} \right) \right]_{0}^{\infty}$

Now, evaluate the limits:

1. As $x \to \infty$, $\tan^{-1} \left( \frac{x}{3} \right) \to \frac{\pi}{2}$.
2. At $x = 0$, $\tan^{-1} \left( \frac{0}{3} \right) = 0$.

Therefore:

$\frac{2}{3} \left[ \frac{\pi}{2} - 0 \right] = \frac{2}{3} \times \frac{\pi}{2} = \frac{\pi}{3}$

Conclusion:

The value of the integral is $\frac{\pi}{3}$.

Thus, the correct answer is:

$\boxed{\frac{\pi}{3}}$

Would you like further details on any part of the process or have questions?

Here are 5 related questions to explore further:

1. How do improper integrals work and why are limits needed for infinity?
2. What is the derivation of the arctangent integration formula?
3. How would the integral change if the denominator was $x^2 - 9$ instead of $9 + x^2$?
4. Can improper integrals sometimes be divergent? How do we recognize such cases?
5. What are some other types of integrals that can be solved using trigonometric substitution?

Tip: When evaluating improper integrals, always check if the function tends to infinity or remains bounded as the limit goes to infinity to determine if the integral is convergent or divergent.