To evaluate the integral \(\int_{1}^{\infty} \frac{\ln(x)}{x} \, dx\), we'll go through the following steps:

1. **Integral Setup:**
   \[
   I = \int_{1}^{\infty} \frac{\ln(x)}{x} \, dx
   \]

2. **Substitution:**
   Let \( u = \ln(x) \), so \( du = \frac{1}{x} \, dx \).

   When \( x = 1 \), \( u = \ln(1) = 0 \).

   When \( x \to \infty \), \( u \to \infty \).

   The integral now becomes:
   \[
   I = \int_{0}^{\infty} u \, du
   \]

3. **Evaluate the Integral:**
   The integral of \( u \) with respect to \( u \) is straightforward:
   \[
   I = \left. \frac{u^2}{2} \right|_{0}^{\infty}
   \]

   Evaluating this:
   \[
   I = \lim_{u \to \infty} \frac{u^2}{2} - \frac{0^2}{2} = \infty
   \]

The integral diverges, meaning it does not converge to a finite value. Therefore:

\[
\int_{1}^{\infty} \frac{\ln(x)}{x} \, dx \text{ diverges.}
\]

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What is an example of a similar integral that converges?
2. How can you determine whether an improper integral converges?
3. What is the integral of \(\frac{1}{x} \ln(x)\) over a finite interval, such as \([1, e]\)?
4. How does substitution affect the bounds of integration?
5. Can we use any other techniques, like integration by parts, to solve this integral?

**Tip:** When dealing with improper integrals, always check for divergence by evaluating limits, especially as the variable approaches infinity or a point of discontinuity.

Learn how to evaluate the improper integral of (lnx)/x from 1 to infinity using the substitution method. This problem explores calculus concepts related to improper integrals and substitution techniques. Suitable for undergraduate students and anyone interested in advanced calculus topics.

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