https://r2.erweima.ai/mathbot/image/1725568613185q4hgn4ra.jpg

The integral shown in the image is:

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

### Step-by-step solution:

1. **Simplify the expression**:
   \[
   \ln(x^3) = 3 \ln(x)
   \]
   Thus, the integral becomes:
   \[
   \int \frac{1}{x \cdot 3 \ln(x)} \, dx = \frac{1}{3} \int \frac{1}{x \ln(x)} \, dx
   \]

2. **Substitute**:
   Let \( u = \ln(x) \), so that \( du = \frac{1}{x} \, dx \).

   The integral transforms into:
   \[
   \frac{1}{3} \int \frac{1}{u} \, du
   \]

3. **Integrate**:
   The integral of \( \frac{1}{u} \) is \( \ln|u| \), so:
   \[
   \frac{1}{3} \ln|\ln(x)| + C
   \]

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

### Final answer:
\[
\int \frac{1}{x \ln(x^3)} \, dx = \frac{1}{3} \ln|\ln(x)| + C
\]

Would you like more details or have any questions? Here are 5 related questions for practice:

1. What is the general method for solving integrals involving logarithms?
2. How does the chain rule apply when differentiating logarithmic functions?
3. What would the integral of \( \frac{1}{x \ln(x^n)} \) be for any positive integer \( n \)?
4. How do you handle logarithmic integrals that include powers of \( x \)?
5. Can this method be extended to integrals involving \( \ln(x^k) \) for any constant \( k \)?

**Tip:** Whenever you encounter a logarithmic term in the denominator, substitution is often the key to simplifying the integral.

Solve the integral \( \int \frac{1}{x \ln(x^3)} \, dx \)

Learn how to solve the integral \( \int \frac{1}{x \ln(x^3)} \, dx \) step-by-step using substitution. This calculus problem covers key concepts such as logarithmic functions and substitution methods in integrals. Suitable for Grades 11-12 and college-level students.

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