This is a question about determining the convergence or divergence of an improper integral using comparison tests. The integral in question is:

$\int_1^\infty \frac{8 + 3 \cos x}{2 + 3x^2} \, dx$

The problem is asking you to choose the correct reasoning for convergence or divergence by comparing the given function to simpler functions. Let’s break down the key ideas:

1. Comparison Test:

   • If $f(x) \leq g(x)$ for all $x \geq a$ and $\int_a^\infty g(x) \, dx$ converges, then $\int_a^\infty f(x) \, dx$ also converges.
   • If $f(x) \geq g(x)$ for all $x \geq a$ and $\int_a^\infty g(x) \, dx$ diverges, then $\int_a^\infty f(x) \, dx$ also diverges.

2. Behavior of the Function:

   • For large $x$, the dominant term in the denominator $2 + 3x^2$ is $3x^2$. Hence, the given function behaves like $\frac{8}{3x^2}$ asymptotically.
   • Using the comparison test, you compare $\frac{8 + 3\cos x}{2 + 3x^2}$ to simpler bounding functions such as $\frac{8}{3x^2}$, $\frac{11}{5x^2}$, or $\frac{5}{3x^2}$, based on the options provided.

3. Evaluation of Simplified Integral:

   • The integral of $\frac{c}{x^2}$ for $x \geq 1$ is: $\int_1^\infty \frac{c}{x^2} \, dx = c \int_1^\infty x^{-2} \, dx = c \left[ -\frac{1}{x} \right]_1^\infty = c \cdot \frac{1}{1} = c.$
   • Since the result is finite, $\int_1^\infty \frac{c}{x^2} \, dx$ converges for any constant $c > 0$.

4. Correct Choice:

   • From the given options, the correct choice is: $\text{The integral is convergent since } \frac{8 + 3 \cos x}{2 + 3x^2} \leq \frac{8}{3x^2} \text{ for all } x \geq 1 \text{ and } \int_1^\infty \frac{8}{3x^2} \, dx = \frac{8}{3}.$

This matches the second option in the list. Therefore, the second choice is correct.

Would you like further clarification or details?

Related Questions:

1. What is the difference between convergence and divergence of an improper integral?
2. How does the comparison test help in analyzing improper integrals?
3. Why does the integral of $\frac{1}{x^2}$ converge, while the integral of $\frac{1}{x}$ diverges?
4. How do you determine dominant terms for asymptotic behavior of functions?
5. What other methods can be used to test the convergence of improper integrals?

Tip: For comparison tests, always simplify the dominant terms of the numerator and denominator to find an easy comparison function.