The question asks us to determine if the improper integral

$I = \int_0^1 \frac{dx}{\sqrt{25x^3 + 4x^5}}$

is convergent using a comparison test. If convergent, we are to provide an upper bound for its value. Let us analyze the given multiple-choice options systematically.

Step-by-Step Analysis:

1. Behavior of the Denominator:

   • The denominator is $\sqrt{25x^3 + 4x^5}$. For small $x$ (near $x = 0$), the term $25x^3$ dominates, while for larger $x$ (closer to $x = 1$), both $25x^3$ and $4x^5$ contribute significantly.

2. Comparison Test Setup:

   • To use the comparison test, we need bounding functions $g(x)$ such that: $g(x) \leq \frac{1}{\sqrt{25x^3 + 4x^5}} \leq h(x)$ with $g(x)$ and $h(x)$ being simpler expressions that allow for straightforward evaluation.

3. Analysis of Options:

   • Option (a): Compares the denominator to $4x^5$, which overestimates the denominator significantly for $x \to 0$, hence underestimates the integral. Incorrect.

   • Option (b): Suggests comparing with $\sqrt{29x^5}$, which is incorrect as it assumes dominance of $x^5$ for all $x$, even near 0. This fails for small $x$.

   • Option (c): Correctly compares the denominator to $\sqrt{25x^3}$ for small $x$ and bounds the integral with $\frac{1}{x^{3/2}}$. The computation of the upper bound is also correctly evaluated: $\int_0^1 \frac{1}{x^{3/2}} dx = 2.$

   • Option (d): Claims divergence using an incorrect comparison with $\sqrt{29x^3}$, which underestimates the denominator for $x \to 0$.

Correct Answer:

The correct reasoning is (c).

Would you like a deeper explanation of any specific step or the reasoning behind eliminating the incorrect choices?

Further Questions:

1. How does the comparison test work for improper integrals?
2. Why does the integral converge for the given bounds in option (c)?
3. Could other bounding functions (e.g., higher-order approximations) simplify this evaluation?
4. What happens if we were to integrate without simplifying the denominator?
5. What are practical uses of determining the convergence of improper integrals?

Tip:

When performing comparison tests for convergence, always analyze the dominant terms of the integrand as $x$ approaches the bounds of integration.