To solve the integral \(\int x \ln(x) \, dx\), we can use integration by parts. 

Integration by parts formula is given by:

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

For the integral \(\int x \ln(x) \, dx\), we will choose:

- \( u = \ln(x) \), hence \( du = \frac{1}{x} dx \)
- \( dv = x \, dx \), hence \( v = \frac{x^2}{2} \)

Now, applying the integration by parts formula:

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

Simplify the integral:

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

Now, integrate \(\int x \, dx\):

\[
\int x \, dx = \frac{x^2}{2}
\]

Thus, the expression becomes:

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

Simplify further:

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

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

So, the final answer is:

\[
\boxed{\int x \ln(x) \, dx = \frac{x^2}{2} \ln(x) - \frac{x^2}{4} + C}
\]

Would you like any further explanation or details on any part of the solution?

Here are five related questions for further practice:
1. What is the integral \(\int \ln(x) \, dx\)?
2. How would you solve the integral \(\int x^2 \ln(x) \, dx\)?
3. Can you solve the integral \(\int x \ln(x^2) \, dx\)?
4. What is the result of the integral \(\int x \ln(x) \, d(x^2)\)?
5. How would you evaluate the integral \(\int x^2 e^x \, dx\)?

**Tip:** When using integration by parts, always carefully choose \( u \) and \( dv \) to simplify the integral.

Learn how to solve the integral ∫ x ln(x) dx using integration by parts. This problem involves advanced calculus techniques and logarithmic functions. Suitable for advanced students and enthusiasts of calculus.

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