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We are given the integral

\[
\int \frac{\ln(9x)}{x^3} \, dx
\]

and we need to apply integration by parts, choosing \( u = \ln(9x) \). In this method, we need to find the functions \( F(x) \) and \( G(x) \) as indicated.

### Step 1: Assign parts for integration by parts
We know that for integration by parts:

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

Here, we let:
- \( u = \ln(9x) \), which implies \( du = \frac{1}{x} \, dx \)
- \( dv = \frac{1}{x^3} \, dx \), so we need to find \( v \).

### Step 2: Integrate \( dv \) to find \( v \)
We can integrate \( dv = \frac{1}{x^3} \, dx \) as follows:

\[
v = \int \frac{1}{x^3} \, dx = -\frac{1}{2x^2}
\]

### Step 3: Apply the integration by parts formula
Now, substitute into the integration by parts formula:

\[
\int \frac{\ln(9x)}{x^3} \, dx = \ln(9x) \left( -\frac{1}{2x^2} \right) - \int \left( -\frac{1}{2x^2} \cdot \frac{1}{x} \right) \, dx
\]

Simplify the integrand in the remaining integral:

\[
= -\frac{\ln(9x)}{2x^2} + \frac{1}{2} \int \frac{1}{x^3} \, dx
\]

### Step 4: Complete the calculation
The second integral is:

\[
\int \frac{1}{x^3} \, dx = -\frac{1}{2x^2}
\]

Thus, the final expression is:

\[
-\frac{\ln(9x)}{2x^2} + \frac{1}{4x^2} + C
\]

### Conclusion:
The function \( F(x) \) is the first term obtained from the integration by parts:

\[
F(x) = -\frac{\ln(9x)}{2x^2}
\]

The function \( G(x) \) is the integrand from the second integral:

\[
G(x) = \frac{1}{x^3}
\]

So, we have:
- \( F(x) = -\frac{\ln(9x)}{2x^2} \)
- \( G(x) = \frac{1}{x^3} \)

Would you like any further clarifications or details? Here are five follow-up questions to deepen your understanding:

1. How is the formula for integration by parts derived?
2. Why was \( u = \ln(9x) \) chosen in this problem?
3. How do you determine which term to set as \( u \) in integration by parts?
4. What would happen if we chose \( u \) and \( dv \) differently in this problem?
5. How would you solve this integral using substitution or another method?

**Tip:** When using integration by parts, always choose \( u \) as the function that simplifies upon differentiation.

Consider the integral \( \int \frac{\ln(9x)}{x^3} \, dx \). After applying the integration by parts with \( u = \ln(9x) \), the original integral is transformed into \( F(x) + \int G(x) \, dx \). Determine the functions \( F(x) \) and \( G(x) \).

Learn how to solve the integral \( \int \frac{\ln(9x)}{x^3} \, dx \) using the integration by parts technique. The solution involves applying the formula \( \int u \, dv = u v - \int v \, du \), assigning \( u = \ln(9x) \) and solving step by step. This problem is ideal for undergraduate students in calculus.

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