The integral you're asking to solve is:

\[
\int \frac{6 \ln(x)}{x} \, dx
\]

This can be solved using the technique of integration by parts. The formula for integration by parts is:

\[
\int u \, dv = uv - \int v \, du
\]

### Step 1: Identify \( u \) and \( dv \)

We choose:
- \( u = \ln(x) \) (since the derivative of \( \ln(x) \) simplifies nicely)
- \( dv = \frac{6}{x} \, dx \)

### Step 2: Differentiate \( u \) and integrate \( dv \)

- \( du = \frac{1}{x} \, dx \)
- \( v = 6 \int \frac{1}{x} \, dx = 6 \ln(x) \)

### Step 3: Apply the integration by parts formula

Now apply the formula:

\[
\int \frac{6 \ln(x)}{x} \, dx = 6 \ln(x) \cdot \ln(x) - \int 6 \ln(x) \cdot \frac{1}{x} \, dx
\]

### Step 4: Simplify the resulting expression

\[
\int \frac{6 \ln(x)}{x} \, dx = 6 (\ln(x))^2 - \int 6 \cdot \frac{1}{x} \, dx
\]

The remaining integral is straightforward:

\[
\int \frac{6}{x} \, dx = 6 \ln(x)
\]

### Step 5: Final answer

Thus, the final result is:

\[
\int \frac{6 \ln(x)}{x} \, dx = 6 (\ln(x))^2 - 6 \ln(x) + C
\]

where \( C \) is the constant of integration.

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Would you like further details on integration by parts, or do you have any other questions?

Here are some related questions for practice:
1. Compute \( \int \frac{\ln(x)}{x^2} \, dx \).
2. Solve \( \int x \ln(x) \, dx \) using integration by parts.
3. Find \( \int \ln^2(x) \, dx \).
4. Evaluate \( \int e^x \ln(x) \, dx \).
5. Compute \( \int \frac{\ln(x)}{x} \, dx \).

**Tip**: When choosing \( u \) in integration by parts, logarithmic functions often simplify well as \( u \) since their derivatives are simpler.

Learn how to solve the integral of 6ln(x)/x step-by-step using integration by parts. This calculus problem covers important concepts like logarithmic functions and integration by parts, and is suitable for advanced high school or college-level students.

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