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The given integral is:

\[
\int x^4 \ln{x} \, dx
\]

We can solve this using **integration by parts**, which is a method where we express the integral of a product of two functions as follows:

\[
\int u \, dv = u v - \int v \, du
\]

For this integral, we can choose:
- \( u = \ln{x} \), because its derivative simplifies the problem,
- \( dv = x^4 dx \), since the integral of \( x^4 \) is straightforward.

### Step 1: Compute \( du \) and \( v \)
- \( u = \ln{x} \Rightarrow du = \frac{1}{x} dx \)
- \( dv = x^4 dx \Rightarrow v = \frac{x^5}{5} \)

### Step 2: Apply the Integration by Parts Formula
Now, apply the formula \( \int u \, dv = u v - \int v \, du \):

\[
\int x^4 \ln{x} \, dx = \ln{x} \cdot \frac{x^5}{5} - \int \frac{x^5}{5} \cdot \frac{1}{x} dx
\]

### Step 3: Simplify the Remaining Integral
Simplify the expression:

\[
= \frac{x^5 \ln{x}}{5} - \frac{1}{5} \int x^4 dx
\]

The remaining integral is:

\[
\int x^4 dx = \frac{x^5}{5}
\]

### Step 4: Final Expression
Substitute this result back into the equation:

\[
\frac{x^5 \ln{x}}{5} - \frac{1}{5} \cdot \frac{x^5}{5} = \frac{x^5 \ln{x}}{5} - \frac{x^5}{25}
\]

Thus, the solution to the integral is:

\[
\boxed{\frac{x^5 \ln{x}}{5} - \frac{x^5}{25} + C}
\]

where \( C \) is the constant of integration.

---

Would you like a more detailed breakdown or clarification on any part? Here are some related questions to explore:

1. How does integration by parts work in other scenarios?
2. How can we apply substitution for integrals involving logarithmic functions?
3. What is the general process for solving integrals involving products of functions?
4. Can we solve this integral using a different method, like substitution?
5. What happens if we integrate functions involving higher powers of \( x \) and \( \ln{x} \)?

**Tip**: When using integration by parts, always choose \( u \) as the function that simplifies upon differentiation (like \( \ln{x} \) here).

Learn how to compute the integral ∫ x^4 ln(x) dx using the method of integration by parts. This solution breaks down the steps, including choosing the correct u and dv, and provides a detailed explanation suitable for university-level calculus students.

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