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?

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.