The integral you provided appears to be:

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

This is an integration problem that can be solved using integration by parts. Here's the general process to solve this:

Step 1: Integration by parts formula

The formula for integration by parts is:

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

We need to choose $u$ and $dv$ appropriately. For this problem, we can choose:

• $u = \ln(x)$, so $du = \frac{1}{x} dx$
• $dv = x^p dx$, so $v = \frac{x^{p+1}}{p+1}$ (assuming $p \neq -1$)

Step 2: Apply the formula

Using the integration by parts formula:

$\int x^p \ln(x) \, dx = \ln(x) \cdot \frac{x^{p+1}}{p+1} - \int \frac{x^{p+1}}{p+1} \cdot \frac{1}{x} \, dx$

Simplify the second term:

$\int \frac{x^{p+1}}{p+1} \cdot \frac{1}{x} \, dx = \frac{1}{p+1} \int x^p \, dx$

The integral of $x^p$ is:

$\int x^p \, dx = \frac{x^{p+1}}{p+1}$

Step 3: Final result

Substitute this back:

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

where $C$ is the constant of integration.

Summary

The solution to the integral is:

$\int x^p \ln(x) \, dx = \frac{x^{p+1}}{p+1} \ln(x) - \frac{x^{p+1}}{(p+1)^2} + C$

Do you need further clarification on any part of this process?

Here are five questions to help expand your understanding:

1. How would the process change if $p = -1$?
2. Can you apply integration by parts to a different function, such as $x^p e^x$?
3. What happens if we choose a different $u$ and $dv$ in integration by parts?
4. How would you approach this problem if asked to solve it using a different method, such as substitution?
5. What are the conditions under which integration by parts is most useful?

Tip: When applying integration by parts, always choose $u$ as the function that simplifies when differentiated and $dv$ as the function that can be easily integrated.