To solve the integral

$\int \frac{1}{x \ln(x)} \, dx,$

we can proceed as follows:

Step 1: Substitution

Let $u = \ln(x)$, which implies that $du = \frac{1}{x} dx$.

Step 2: Rewrite the Integral

Substituting $u$ into the integral, we have:

$\int \frac{1}{x \ln(x)} \, dx = \int \frac{1}{u} \, du.$

Step 3: Solve the Integral

The integral $\int \frac{1}{u} \, du$ is a standard integral:

$\int \frac{1}{u} \, du = \ln|u| + C.$

Step 4: Substitute Back $u = \ln(x)$

Substitute $u = \ln(x)$ back into the solution:

$\ln|u| + C = \ln|\ln(x)| + C.$

Final Answer:

$\int \frac{1}{x \ln(x)} \, dx = \ln|\ln(x)| + C.$

Would you like a deeper explanation of any step or examples of similar integrals?

Here are some related questions for you to consider:

1. What if the integral had a different denominator, such as $x (\ln(x))^2$?
2. Can this integral appear in real-life applications? If so, where?
3. What are some alternative methods to solve this integral?
4. How does this technique extend to integrals involving functions like $\ln(x)$ or $\ln(\ln(x))$?
5. How can improper integrals involving $\frac{1}{x \ln(x)}$ be handled over different intervals?

Tip: Always check whether a substitution simplifies the integral—it is often a key technique in solving complicated expressions.