It seems like there's a typo in your question. I'll assume you meant to find the integral of \( \frac{1}{x (\ln x)^2} \, dx \).

To solve this, we can use substitution.

Let:
\[
u = \ln x
\]
Then, differentiate \( u \) with respect to \( x \):
\[
\frac{du}{dx} = \frac{1}{x} \quad \text{or} \quad du = \frac{dx}{x}
\]

Now, substitute into the integral:
\[
\int \frac{1}{x (\ln x)^2} \, dx = \int \frac{1}{u^2} \, du
\]

The integral of \( \frac{1}{u^2} \) with respect to \( u \) is:
\[
\int \frac{1}{u^2} \, du = -\frac{1}{u} + C
\]

Substituting back \( u = \ln x \), we get:
\[
\int \frac{1}{x (\ln x)^2} \, dx = -\frac{1}{\ln x} + C
\]
where \( C \) is the constant of integration.

Would you like more details or have any questions?

Here are some related questions:
1. How can substitution be used for more complex logarithmic integrals?
2. What is the significance of logarithmic functions in calculus?
3. How do you approach integrals involving powers of logarithmic functions?
4. Can this integral be solved using other methods, such as integration by parts?
5. How does this result apply to real-world problems involving logarithmic functions?

**Tip:** When dealing with integrals involving logarithms, substitution is often a useful technique to simplify the expression.

Learn how to integrate the function 1 / (x (ln x)^2) dx using the substitution method. Step-by-step solution with detailed explanations and examples. Suitable for college-level calculus students.

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