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We are tasked with determining if the improper integral

\[
I = \int_0^1 \frac{2 \ln(6x)}{\sqrt{x}} \, dx
\]

converges or diverges. Let's proceed step by step.

### Step 1: Recognizing the improper nature
The integral is improper because of the term \(\frac{1}{\sqrt{x}}\) at \(x = 0\), which implies a singularity at \(x = 0\). We need to check the behavior of the integrand as \(x \to 0\).

### Step 2: Analyzing the integrand
We can split the integrand into:

\[
\frac{2 \ln(6x)}{\sqrt{x}} = 2 \frac{\ln(6x)}{x^{1/2}}
\]

- As \(x \to 1\), \(\ln(6x)\) remains finite, so there is no problem at the upper limit.
- As \(x \to 0\), \(\ln(6x)\) behaves like \(\ln(x)\), which goes to \(-\infty\). Thus, the behavior near \(x = 0\) is crucial to determining convergence.

### Step 3: Approximation near \(x = 0\)
As \(x \to 0\), \(\ln(6x) \sim \ln(x)\), and we approximate the integrand by:

\[
\frac{2 \ln(x)}{\sqrt{x}}
\]

### Step 4: Estimating the integral near 0
Now we check the behavior of the integral near \(x = 0\) by considering:

\[
\int_0^\epsilon \frac{\ln(x)}{x^{1/2}} \, dx
\]

This is a standard improper integral. We use integration by parts, or we can check its convergence by knowing that integrals of the form \(\int_0^\epsilon x^a \ln(x) \, dx\) converge for certain values of \(a\).

- For the term \(\frac{\ln(x)}{\sqrt{x}}\), the integral will diverge as \(x \to 0\), since \(\ln(x)\) grows negatively unbounded while \(x^{-1/2}\) blows up at \(x = 0\).

### Step 5: Conclusion
Thus, the integral diverges near \(x = 0\) because the singularity in \(\frac{\ln(x)}{\sqrt{x}}\) leads to a non-convergent behavior.

### Final Answer:
The improper integral \(I = \int_0^1 \frac{2 \ln(6x)}{\sqrt{x}} \, dx\) **diverges**.

Would you like any further details or explanations?

Here are 5 related questions for further exploration:
1. What conditions are necessary for an improper integral to converge at a singular point?
2. Can the integral \(\int_0^1 \frac{1}{\sqrt{x}} \, dx\) converge, and why?
3. How does the presence of \(\ln(x)\) affect the convergence of integrals at 0?
4. Can we change the limits or substitution to handle the singularity?
5. What are other methods for evaluating improper integrals?

**Tip:** When handling improper integrals, always check the limits where the integrand may become undefined or infinite.

Determine if the improper integral I = ∫₀¹ (2 ln(6x) / √x) dx converges or diverges.

This problem involves analyzing the convergence or divergence of the improper integral I = ∫₀¹ (2 ln(6x) / √x) dx. It discusses the singularity at x = 0, the logarithmic behavior of ln(x), and evaluates the integral through theorems related to improper integrals.

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