To solve the integral \(\int 4x \cdot 2 \ln(x) \, dx\), we can simplify it first and then use integration by parts.

### Simplification
The integral can be rewritten as:

\[
\int 8x \ln(x) \, dx
\]

### Integration by Parts
For the integral \(\int 8x \ln(x) \, dx\), we'll apply the integration by parts formula:

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

Let's choose:
- \( u = \ln(x) \) (so that \( du = \frac{1}{x} dx \))
- \( dv = 8x \, dx \) (so that \( v = 4x^2 \))

Now, apply the integration by parts formula:

\[
\int 8x \ln(x) \, dx = 4x^2 \ln(x) - \int 4x^2 \cdot \frac{1}{x} \, dx
\]

Simplify the integral:

\[
\int 8x \ln(x) \, dx = 4x^2 \ln(x) - \int 4x \, dx
\]

Now, integrate \(4x\):

\[
\int 4x \, dx = 2x^2
\]

So the expression becomes:

\[
\int 8x \ln(x) \, dx = 4x^2 \ln(x) - 2x^2 + C
\]

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

### Final Answer

\[
\int 4x \cdot 2 \ln(x) \, dx = 4x^2 \ln(x) - 2x^2 + C
\]

Would you like further explanation or have any questions?

Here are 5 related questions:
1. How would you approach \(\int x^2 \ln(x) \, dx\)?
2. What is the integral of \(x \ln(x^2) \, dx\)?
3. Can you solve \(\int \ln(x) \, dx\) without using integration by parts?
4. How would the integral change if it were \(\int 8x \ln(ax) \, dx\) where \(a\) is a constant?
5. What is the integral of \(x \ln(x) \, dx\)?

**Tip:** When dealing with logarithmic functions in integrals, integration by parts is often a useful technique to simplify the problem.

Learn how to solve the integral \( \int 4x^2 \ln(x) \, dx \) step-by-step with detailed explanations. This problem involves integration techniques such as integration by parts and is suitable for advanced calculus students.

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