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.