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To solve this problem, let's go through each step of integration by parts.

### Given Integral
We need to evaluate:
\[
\int \frac{\ln(x)}{x^5} \, dx
\]

### Step 1: Choosing \( f(x) \) and \( g'(x) \)
For integration by parts, we need to choose parts \( f(x) \) and \( g'(x) \) such that:
\[
\int f(x) g'(x) \, dx = f(x) g(x) - \int f'(x) g(x) \, dx
\]

In this case, a good choice is:
- \( f(x) = \ln(x) \) (since the derivative of \( \ln(x) \) is simpler, and it will reduce the complexity of the integral)
- \( g'(x) = \frac{1}{x^5} \)

### Step 2: Finding \( f'(x) \) and \( g(x) \)
1. **Differentiate \( f(x) = \ln(x) \):**
   \[
   f'(x) = \frac{1}{x}
   \]

2. **Integrate \( g'(x) = \frac{1}{x^5} \):**
   \[
   g(x) = \int \frac{1}{x^5} \, dx = \int x^{-5} \, dx = \frac{x^{-4}}{-4} = -\frac{1}{4x^4}
   \]

So we have:
- \( f(x) = \ln(x) \)
- \( g'(x) = \frac{1}{x^5} \)
- \( f'(x) = \frac{1}{x} \)
- \( g(x) = -\frac{1}{4x^4} \)

### Step 3: Applying Integration by Parts
Now, using the integration by parts formula:
\[
\int \frac{\ln(x)}{x^5} \, dx = f(x) g(x) - \int f'(x) g(x) \, dx
\]
Substitute the values we found:
\[
= \ln(x) \cdot \left(-\frac{1}{4x^4}\right) - \int \frac{1}{x} \cdot \left(-\frac{1}{4x^4}\right) \, dx
\]
\[
= -\frac{\ln(x)}{4x^4} + \int \frac{1}{4x^5} \, dx
\]

### Step 4: Simplifying the Integral
Now, integrate \( \frac{1}{4x^5} \):
\[
\int \frac{1}{4x^5} \, dx = \frac{1}{4} \int x^{-5} \, dx = \frac{1}{4} \cdot \frac{x^{-4}}{-4} = -\frac{1}{16x^4}
\]

### Final Answer
Combine the terms:
\[
\int \frac{\ln(x)}{x^5} \, dx = -\frac{\ln(x)}{4x^4} - \frac{1}{16x^4} + C
\]

### Summary
The evaluated integral is:
\[
\int \frac{\ln(x)}{x^5} \, dx = -\frac{\ln(x)}{4x^4} - \frac{1}{16x^4} + C
\]

Would you like a detailed explanation of any step or have further questions? 

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### Relative Questions
1. How do you determine the best choice for \( f(x) \) and \( g'(x) \) in integration by parts?
2. Why is integration by parts useful for integrals involving logarithmic functions?
3. What would change if the integral were \( \int \frac{\ln(x)}{x^4} \, dx \) instead?
4. Can we use substitution in conjunction with integration by parts, and if so, how?
5. How does the choice of \( f(x) \) and \( g(x) \) affect the complexity of the solution?

### Tip
When choosing parts for integration by parts, it’s generally helpful to pick \( f(x) \) as a function that becomes simpler when differentiated (like logarithmic or polynomial functions).

Use integration by parts to evaluate the indefinite integral \( \int \frac{\ln(x)}{x^5} \, dx \).

Learn how to evaluate the indefinite integral \( \int \frac{\ln(x)}{x^5} \, dx \) using integration by parts. This solution covers choosing the right parts, simplifying with logarithmic and power functions, and finding the antiderivative. Suitable for college-level calculus students.

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