We are tasked with solving the integral:

\[
\int_{1}^{2} 12x^2 \ln(x) \, dx
\]

To solve this, we'll use **integration by parts**, which is generally applicable for integrals involving products of functions (in this case, \( x^2 \) and \( \ln(x) \)).

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

We'll choose:
\[
u = \ln(x) \quad \text{so that} \quad du = \frac{1}{x} \, dx
\]
\[
dv = 12x^2 \, dx \quad \text{so that} \quad v = \int 12x^2 \, dx = 4x^3
\]

### Step 2: Apply the integration by parts formula
The formula is:
\[
\int u \, dv = uv - \int v \, du
\]
Substituting \( u \), \( du \), \( v \), and \( dv \):
\[
\int 12x^2 \ln(x) \, dx = 4x^3 \ln(x) - \int 4x^3 \cdot \frac{1}{x} \, dx
\]

Simplify the remaining integral:
\[
\int 4x^3 \cdot \frac{1}{x} \, dx = \int 4x^2 \, dx = \frac{4x^3}{3}
\]

### Step 3: Substitute back
Now we have:
\[
\int 12x^2 \ln(x) \, dx = 4x^3 \ln(x) - \frac{4x^3}{3}
\]

### Step 4: Evaluate the definite integral
We now evaluate this expression from 1 to 2:
\[
\left[ 4x^3 \ln(x) - \frac{4x^3}{3} \right]_{1}^{2}
\]

First, plug in \( x = 2 \):
\[
4(2^3) \ln(2) - \frac{4(2^3)}{3} = 32 \ln(2) - \frac{32}{3}
\]

Then, plug in \( x = 1 \):
\[
4(1^3) \ln(1) - \frac{4(1^3)}{3} = 0 - \frac{4}{3}
\]

### Step 5: Compute the final value
Now subtract the two results:
\[
\left( 32 \ln(2) - \frac{32}{3} \right) - \left( 0 - \frac{4}{3} \right) = 32 \ln(2) - \frac{32}{3} + \frac{4}{3}
\]
\[
= 32 \ln(2) - \frac{28}{3}
\]

Thus, the final result of the definite integral is:
\[
\boxed{32 \ln(2) - \frac{28}{3}}
\]

Would you like any further details or clarifications on this solution?

Here are 5 related questions for you:
1. How do we choose \( u \) and \( dv \) in integration by parts?
2. Can integration by parts be applied more than once in a problem?
3. How would the result change if the limits of integration were different?
4. What other methods could be used to solve this integral, if any?
5. How does the logarithmic function affect the complexity of integration?

**Tip:** When solving integrals, knowing the properties of basic functions like \( \ln(x) \), \( e^x \), and polynomials simplifies the process of choosing techniques like integration by parts.

Learn how to solve the definite integral of 12x^2 ln(x) from 1 to 2 using the integration by parts technique. This detailed step-by-step solution covers logarithmic functions, algebraic manipulation, and the evaluation of definite integrals. Ideal for students in Grades 11-12 and undergraduate calculus courses.

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