To solve the integral \(\int x^5 \ln(x) \, dx\), we will use the method of integration by parts. 

### Integration by Parts Formula:
\[
\int u \, dv = uv - \int v \, du
\]
Here, we need to choose \( u \) and \( dv \) from the integrand \( x^5 \ln(x) \, dx \).

### Step 1: Choose \( u \) and \( dv \)
Let:
\[
u = \ln(x) \quad \text{and} \quad dv = x^5 \, dx
\]

### Step 2: Compute \( du \) and \( v \)
Differentiate \( u \) to find \( du \):
\[
du = \frac{1}{x} \, dx
\]
Integrate \( dv \) to find \( v \):
\[
v = \frac{x^6}{6}
\]

### Step 3: Apply the Integration by Parts Formula
Now, substitute into the integration by parts formula:
\[
\int x^5 \ln(x) \, dx = \frac{x^6}{6} \ln(x) - \int \left(\frac{x^6}{6} \cdot \frac{1}{x}\right) dx
\]
Simplify the integrand of the remaining integral:
\[
\int x^5 \ln(x) \, dx = \frac{x^6}{6} \ln(x) - \frac{1}{6} \int x^5 \, dx
\]

### Step 4: Integrate the Remaining Integral
Now, integrate \( \int x^5 \, dx \):
\[
\int x^5 \, dx = \frac{x^6}{6}
\]
So, the integral becomes:
\[
\int x^5 \ln(x) \, dx = \frac{x^6}{6} \ln(x) - \frac{1}{6} \cdot \frac{x^6}{6} + C
\]
\[
\int x^5 \ln(x) \, dx = \frac{x^6}{6} \ln(x) - \frac{x^6}{36} + C
\]

### Final Answer
\[
\int x^5 \ln(x) \, dx = \frac{x^6}{6} \ln(x) - \frac{x^6}{36} + C
\]

Do you want any further details or have any questions?

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Here are 5 questions to expand on this concept:
1. How does the choice of \( u \) and \( dv \) affect the integration by parts process?
2. Can you solve \( \int x^2 \ln(x) \, dx \) using a similar method?
3. What happens if you integrate by parts multiple times?
4. How would you handle the integral \( \int x \ln(x) \, dx \)?
5. How can integration by parts be used in solving differential equations?

**Tip:** When choosing \( u \) in integration by parts, it's often useful to pick the function that simplifies when differentiated, such as logarithmic or inverse trigonometric functions.

Learn how to solve the integral \(\int x^5 \ln(x) \, dx\) using integration by parts with a step-by-step solution. This problem involves advanced calculus concepts and logarithmic functions. Suitable for advanced high school and college-level students.

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