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.