We are tasked with finding the integral:

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

To solve this, we can use **integration by parts**. Recall that for integration by parts:

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

### Step 1: Choose \( u \) and \( dv \)
Let:
- \( u = \ln(x) \) (since its derivative simplifies the expression),
- \( dv = \frac{1}{x^3} \, dx = x^{-3} \, dx \).

Now, we calculate \( du \) and \( v \):
- \( du = \frac{1}{x} \, dx \),
- \( v = \int x^{-3} \, dx = \frac{x^{-2}}{-2} = -\frac{1}{2x^2} \).

### Step 2: Apply the formula for integration by parts
Now, substitute into the formula \( \int u \, dv = uv - \int v \, du \):

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

Simplify the first term:

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

### Step 3: Solve the remaining integral
The remaining integral is straightforward:

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

### Step 4: Final expression
Now, substituting this back, we get the final result:

\[
\int \frac{\ln(x)}{x^3} \, dx = -\frac{\ln(x)}{2x^2} - \frac{1}{4x^2} + C
\]

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

Would you like further clarification or details on any of these steps?

Here are 5 related questions:
1. How is integration by parts applied in other problems involving logarithms?
2. What are the conditions for choosing \( u \) and \( dv \) in integration by parts?
3. How do you integrate other functions like \( \ln(x) / x^2 \)?
4. Can integration by parts be applied to trigonometric functions as well?
5. What are the applications of logarithmic integrals in real-world problems?

**Tip:** When using integration by parts, always pick \( u \) as the function that becomes simpler when differentiated.

This page provides a step-by-step solution for integrating ln(x)/x^3 using the integration by parts technique. It covers key calculus concepts and shows how to apply formulas like the power rule and integration by parts. Suitable for undergraduate calculus students.

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