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Let's solve **Question 9**:  
\[
\int x \ln(x) \, dx
\]

### Solution using Integration by Parts:

The formula for **integration by parts** is:
\[
\int u \, dv = u v - \int v \, du
\]
Here, we need to choose \(u\) and \(dv\). We generally let \(u\) be the part that simplifies when differentiated, and \(dv\) be the remaining part that we can integrate.

1. **Choice of \(u\) and \(dv\):**
   - \(u = \ln(x) \quad \Rightarrow \quad du = \frac{1}{x} \, dx\)
   - \(dv = x \, dx \quad \Rightarrow \quad v = \frac{x^2}{2}\)

2. **Apply the Integration by Parts Formula:**
   \[
   \int x \ln(x) \, dx = \ln(x) \cdot \frac{x^2}{2} - \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx
   \]
   Simplify the second integral:
   \[
   \int \frac{x^2}{2} \cdot \frac{1}{x} \, dx = \int \frac{x}{2} \, dx
   \]

3. **Evaluate the Remaining Integral:**
   \[
   \int \frac{x}{2} \, dx = \frac{1}{2} \int x \, dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4}
   \]

4. **Combine the Results:**
   \[
   \int x \ln(x) \, dx = \frac{x^2 \ln(x)}{2} - \frac{x^2}{4} + C
   \]
   where \(C\) is the constant of integration.

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

Would you like further clarifications or more examples of similar integrals?

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Here are 5 related problems you could explore:
1. \( \int \ln(x) \, dx \)
2. \( \int x^2 \ln(x) \, dx \)
3. \( \int x e^x \, dx \)
4. \( \int x \cos(x) \, dx \)
5. \( \int \ln(2x) \, dx \)

**Tip:** In general, when you see a product of a polynomial and a logarithmic function, integration by parts is a good strategy to try.

Learn how to solve the integral ∫ x ln(x) dx using integration by parts. This detailed step-by-step solution covers the choice of u and dv, application of the integration by parts formula, and final combination of results. Ideal for college calculus students.

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