To solve the integral $\int \frac{dx}{x \ln x}$, we can use the substitution method.

Step 1: Substitution

Let: $u = \ln x$ Then, the derivative of $u$ with respect to $x$ is: $du = \frac{1}{x} dx$ So, $\frac{1}{x} dx = du$.

Step 2: Rewrite the Integral

Now, substitute into the integral: $\int \frac{dx}{x \ln x} = \int \frac{du}{u}$

Step 3: Solve the Integral

We now recognize that $\int \frac{du}{u}$ is a standard integral: $\int \frac{du}{u} = \ln |u| + C$

Step 4: Back-Substitute

Substitute back $u = \ln x$: $\ln |\ln x| + C$

Thus, the solution is: $\boxed{\ln |\ln x| + C}$

Would you like to go over any step in more detail or have another question related to this topic?

Related Questions:

1. How does the substitution method work in other types of integrals?
2. What are some common integrals involving logarithmic functions?
3. Can this integral be solved using integration by parts?
4. How do we handle definite integrals involving $\ln x$?
5. What are some practical applications of logarithmic integrals?

Tip: When using substitution, always try to simplify the integral to a form you can easily integrate, as we did with $\frac{du}{u}$.