The given problem requires determining whether the improper integral

$\int_{1}^{\infty} \frac{8 + 3 \cos x}{2 + 3x^2} \, dx$

is convergent or divergent by comparison with other functions. Let us analyze the provided options systematically:

Step-by-Step Analysis

Key Observations

1. The numerator $8 + 3\cos x$ oscillates between $8 - 3 = 5$ and $8 + 3 = 11$ because $\cos x \in [-1, 1]$.
2. The denominator $2 + 3x^2$ grows quadratically as $x \to \infty$.
3. The behavior of the integral is dominated by the term $\frac{1}{3x^2}$ for large $x$.

Testing Convergence

We compare $\frac{8 + 3 \cos x}{2 + 3x^2}$ to simpler bounding functions:

1. Upper Bound: Since $5 \leq 8 + 3\cos x \leq 11$, it follows that: $\frac{5}{2 + 3x^2} \leq \frac{8 + 3\cos x}{2 + 3x^2} \leq \frac{11}{2 + 3x^2}.$

2. Comparison with $\frac{k}{x^2}$:

   • For $x \to \infty$, the denominator $2 + 3x^2$ behaves like $3x^2$, leading to approximate bounds: $\frac{5}{3x^2} \leq \frac{8 + 3\cos x}{2 + 3x^2} \leq \frac{11}{3x^2}.$

3. Convergence of Bounds:

   • The integral of $\frac{k}{x^2}$ for $k > 0$ is: $\int_{1}^{\infty} \frac{k}{x^2} \, dx = \left[ -\frac{k}{x} \right]_{1}^{\infty} = k.$
   • Thus, $\int_{1}^{\infty} \frac{5}{3x^2} \, dx = \frac{5}{3} < \infty$ and $\int_{1}^{\infty} \frac{11}{3x^2} \, dx = \frac{11}{3} < \infty$.

4. Conclusion: By the comparison test, the given integral converges because it is bounded above and below by convergent integrals.

Correct Answer

The correct option is (b): $\text{The integral is convergent since } \frac{8 + 3 \cos x}{2 + 3x^2} \leq \frac{8}{3x^2} \text{ for all } x \geq 1 \text{ and } \int_{1}^{\infty} \frac{8}{3x^2} \, dx = \frac{8}{3} < \infty.$

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Related Questions:

1. Why does the denominator $2 + 3x^2$ dominate the numerator as $x \to \infty$?
2. How does the comparison test work for improper integrals?
3. Could we directly evaluate this integral instead of using a comparison?
4. Why do oscillatory terms like $3\cos x$ not affect convergence?
5. What happens if the denominator grows slower, e.g., $2 + 3x$?

Tip: Always simplify the dominant term for large $x$ to assess convergence behavior in improper integrals.