To evaluate the integral

$\int x^2 \ln(x) \, dx,$

we can use the technique of integration by parts. The formula for integration by parts is:

$\int u \, dv = uv - \int v \, du.$

Step 1: Choose $u$ and $dv$

Let's choose: $u = \ln(x), \quad dv = x^2 \, dx.$

Then we compute $du$ and $v$:

$du = \frac{1}{x} \, dx, \quad v = \int x^2 \, dx.$

To find $v$, integrate $x^2$:

$v = \int x^2 \, dx = \frac{x^3}{3}.$

Step 2: Apply the Integration by Parts Formula

Substitute into the integration by parts formula:

$\int x^2 \ln(x) \, dx = u v - \int v \, du.$

Substitute the expressions for $u, v, du,$ and $dv$:

$\int x^2 \ln(x) \, dx = \ln(x) \cdot \frac{x^3}{3} - \int \frac{x^3}{3} \cdot \frac{1}{x} \, dx.$

Simplify the expression:

$\int x^2 \ln(x) \, dx = \frac{x^3 \ln(x)}{3} - \int \frac{x^2}{3} \, dx.$

$\int x^2 \ln(x) \, dx = \frac{x^3 \ln(x)}{3} - \frac{1}{3} \int x^2 \, dx.$

Step 3: Evaluate the Remaining Integral

The remaining integral is $\int x^2 \, dx$:

$\int x^2 \, dx = \frac{x^3}{3}.$

Thus,

$\frac{1}{3} \int x^2 \, dx = \frac{1}{3} \cdot \frac{x^3}{3} = \frac{x^3}{9}.$

Step 4: Combine Results

Substitute this back into the expression:

$\int x^2 \ln(x) \, dx = \frac{x^3 \ln(x)}{3} - \frac{x^3}{9} + C.$

Final Answer

$\int x^2 \ln(x) \, dx = \frac{x^3 \ln(x)}{3} - \frac{x^3}{9} + C.$

This is the evaluated integral using integration by parts.

Would you like more details on any step or have any other questions?

Here are 5 related questions for further practice:

1. Evaluate $\int x \ln(x) \, dx$ using integration by parts.
2. Solve $\int x^2 e^x \, dx$ using integration by parts.
3. Calculate $\int \ln(x) \, dx$ using integration by parts.
4. Determine $\int x^3 \ln(x) \, dx$ using integration by parts.
5. Evaluate $\int x \ln(x^2) \, dx$ using integration by parts.

Tip: When choosing $u$ and $dv$ for integration by parts, pick $u$ to be a function that simplifies upon differentiation, and $dv$ such that it is straightforward to integrate.