To solve the integral \( \int \frac{1}{x (\ln x)^2} \, dx \), we can use substitution.

### Step-by-Step Solution:

1. **Substitute** \( u = \ln x \).  
   Then, \( du = \frac{1}{x} \, dx \), so \( dx = x \, du \).

2. **Rewrite the integral**:
   \[
   \int \frac{1}{x (\ln x)^2} \, dx = \int \frac{1}{u^2} \cdot du
   \]

3. **Integrate** \( \int \frac{1}{u^2} \, du \):
   The integral of \( \frac{1}{u^2} \) is \( -\frac{1}{u} \).

   Therefore:
   \[
   -\frac{1}{u} + C
   \]

4. **Substitute back** \( u = \ln x \):
   \[
   -\frac{1}{\ln x} + C
   \]

### Final Answer:
\[
\int \frac{1}{x (\ln x)^2} \, dx = -\frac{1}{\ln x} + C
\]

Would you like any more details, or do you have any questions?

Here are 5 related questions to further your understanding:

1. How would you solve \( \int \frac{1}{x \ln x} \, dx \)?
2. What is the integral \( \int \frac{1}{(\ln x)^2} \, dx \)?
3. How would the solution change if the integrand were \( \frac{1}{x (\ln x)^3} \)?
4. Can you evaluate \( \int x^2 \ln x \, dx \) using integration by parts?
5. What happens if you try to integrate \( \int \frac{1}{x^2 (\ln x)^2} \, dx \)?

**Tip:** Substitution is a powerful technique when you see a function and its derivative present in the integrand.

Learn how to solve the integral ∫(1 / (x (ln x)^2)) dx using substitution method with a step-by-step explanation. This problem involves calculus concepts and integration techniques, suitable for college-level students.

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