To solve the integral \(\int_{1}^{\infty} \frac{\ln x}{x} \, dx\), let's proceed with the integration step by step.

### Step 1: Recognize the integral form
The given integral is:
\[
\int_{1}^{\infty} \frac{\ln x}{x} \, dx
\]

### Step 2: Perform substitution
Let \( u = \ln x \). Then, the differential \( du = \frac{1}{x} dx \).

This transforms the integral into:
\[
\int_{0}^{\infty} u \, du
\]
(Notice that when \( x = 1 \), \( u = \ln(1) = 0 \), and as \( x \) approaches infinity, \( u \) also approaches infinity.)

### Step 3: Integrate
Now, integrate \( u \) with respect to \( u \):
\[
\int u \, du = \frac{u^2}{2}
\]

### Step 4: Apply the limits
Substitute the limits \( u = 0 \) and \( u \rightarrow \infty \) into the integrated function:
\[
\left[ \frac{u^2}{2} \right]_{0}^{\infty} = \lim_{u \to \infty} \frac{u^2}{2} - \frac{0^2}{2}
\]

As \( u \) approaches infinity, \( \frac{u^2}{2} \) also approaches infinity.

### Conclusion
The integral diverges, meaning:
\[
\int_{1}^{\infty} \frac{\ln x}{x} \, dx = \infty
\]
So, the integral does not converge to a finite value.

Would you like further details on this process or have any questions?

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**Related Questions:**

1. How can we determine if an integral converges or diverges?
2. What are common techniques for solving improper integrals?
3. How does substitution simplify integrals with logarithmic functions?
4. Can you provide an example of a convergent improper integral?
5. What are the applications of improper integrals in physics?

**Tip:** When dealing with improper integrals, always check the behavior of the integrand at the limits of integration to assess convergence.

Learn how to solve the improper integral ∫(ln x)/x from 1 to ∞ with detailed steps and analysis. Understand the concept of improper integrals and their divergence, applying integration by substitution to transform the integral. Suitable for advanced mathematics enthusiasts and students studying calculus.

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