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The integral provided is:

\[
\int \frac{13 \ln(x)}{x \sqrt{1 + (\ln(x))^2}} \, dx
\]

This integral can be solved using a substitution method. Let's proceed step by step.

### Step 1: Substitution
We observe that the integrand involves both \(\ln(x)\) and \(1 + (\ln(x))^2\). This suggests we try a substitution related to \(\ln(x)\).

Let:
\[
u = \ln(x)
\]
Then, differentiate \(u\) with respect to \(x\):
\[
du = \frac{1}{x} \, dx \quad \text{or} \quad dx = x \, du
\]

### Step 2: Simplifying the Integral
Substitute \(u = \ln(x)\) into the integral:
\[
\int \frac{13 u}{x \sqrt{1 + u^2}} \cdot x \, du
\]
The \(x\) terms cancel out:
\[
\int \frac{13 u}{\sqrt{1 + u^2}} \, du
\]

### Step 3: Solving the Integral
Now, the integral becomes:
\[
13 \int \frac{u}{\sqrt{1 + u^2}} \, du
\]

This integral is a standard form, and the solution is:
\[
13 \cdot \sqrt{1 + u^2} + C
\]

### Step 4: Back-Substitute \(u = \ln(x)\)
Finally, substitute \(u = \ln(x)\) back into the solution:
\[
13 \sqrt{1 + (\ln(x))^2} + C
\]

### Final Answer:
\[
\int \frac{13 \ln(x)}{x \sqrt{1 + (\ln(x))^2}} \, dx = 13 \sqrt{1 + (\ln(x))^2} + C
\]

Would you like further explanation on any of the steps? Feel free to ask!

Here are some related questions:
1. What is the general method for solving integrals with logarithmic terms?
2. How do you recognize when to use substitution in integrals?
3. What are some standard integrals involving \(\ln(x)\)?
4. How does the derivative of \(\ln(x)\) help in integral transformations?
5. Can we solve similar integrals using other techniques, like integration by parts?

**Tip:** When dealing with integrals involving \(\ln(x)\), try considering substitution with \(u = \ln(x)\) as it often simplifies the problem.

Evaluate the integral: ∫ (13 ln(x)) / (x √(1 + (ln(x))^2)) dx

Learn how to solve the integral ∫ (13 ln(x)) / (x √(1 + (ln(x))^2)) dx step by step using substitution. This problem covers key concepts in integral calculus, focusing on logarithmic functions and the substitution method.

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