Let's analyze each of these improper integrals using the comparison test to determine if they converge or diverge.

1. $\int_{1}^{\infty} \frac{e^{-x}}{x^2} \, dx$

   • As $x \to \infty$, $e^{-x}$ approaches zero very quickly compared to $x^2$. Thus, $\frac{e^{-x}}{x^2}$ converges faster than $\frac{1}{x^2}$, which is known to converge.
   • By comparison with $\int_{1}^{\infty} \frac{1}{x^2} \, dx$, which converges, we conclude that this integral converges.
   • Answer: $\text{A}$ (Convergent) and $\text{F}$ (Comparison with $\int_{1}^{\infty} \frac{1}{x^2} \, dx$).

2. $\int_{1}^{\infty} \frac{\sqrt{x}}{9 + \sin(x)} \, dx$

   • For large $x$, the function $9 + \sin(x)$ is bounded and does not affect the asymptotic behavior of $\frac{\sqrt{x}}{9 + \sin(x)}$.
   • The integral $\int_{1}^{\infty} \sqrt{x} \, dx$ diverges because $\sqrt{x} = x^{1/2}$ and $\int_{1}^{\infty} x^{1/2} \, dx$ diverges for $p = 1/2 < 1$.
   • By comparison with $\int_{1}^{\infty} \sqrt{x} \, dx$, we conclude that this integral diverges.
   • Answer: $\text{B}$ (Divergent) and $\text{D}$ (Comparison with $\int_{1}^{\infty} \sqrt{x} \, dx$).

3. $\int_{0}^{\pi/2} \frac{\sqrt{x}}{\cos^2(x)} \, dx$

   • As $x \to \frac{\pi}{2}$, $\cos(x) \to 0$, which makes $\cos^2(x)$ approach zero, causing the integrand to grow large.
   • We can compare this to $\int_{0}^{\pi/2} \frac{1}{\cos^2(x)} \, dx$, which diverges.
   • Since $\int_{0}^{\pi/2} \frac{1}{\cos^2(x)} \, dx$ diverges, this integral diverges as well.
   • Answer: $\text{B}$ (Divergent) and $\text{C}$ (Comparison with $\int_{0}^{\pi/2} \frac{1}{\cos^2(x)} \, dx$).

4. $\int_{1}^{\infty} \frac{x^2 + 3}{x^3 + 3} \, dx$

   • For large $x$, the $x^3$ term in the denominator and the $x^2$ term in the numerator dominate, so this behaves asymptotically like $\frac{x^2}{x^3} = \frac{1}{x}$.
   • The integral $\int_{1}^{\infty} \frac{1}{x} \, dx$ diverges (harmonic series), so this integral diverges as well.
   • Answer: $\text{B}$ (Divergent) and $\text{E}$ (Comparison with $\int_{1}^{\infty} \frac{1}{x} \, dx$).

Summary of Answers:

1. AF
2. BD
3. BC
4. BE

Would you like further details on any step, or do you have questions about these solutions?

1. What conditions make a function suitable for the comparison test in improper integrals?
2. How can you determine whether a function's integral converges based on its asymptotic behavior?
3. What are some common divergent integrals used as comparison benchmarks?
4. Why does $\int_{1}^{\infty} \frac{1}{x} \, dx$ diverge, and what does this tell us about other integrals with similar behavior?
5. How does the behavior of $e^{-x}$ affect integrals compared to polynomial terms?

Tip: When applying the comparison test, it's often useful to simplify the behavior of functions by focusing on the dominant terms for large values of $x$ (or near points of discontinuity) to judge convergence or divergence.