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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.

Solve the integral \(\int x^p \ln(x) \, dx\)

Learn how to solve the integral \(\int x^p \ln(x) \, dx\) using the method of integration by parts. This detailed solution covers key concepts like logarithmic functions and integration by parts, suitable for undergraduate calculus students.

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