To integrate the expression \(\frac{\ln(\ln(5x))}{5x}\), we can use a substitution method to simplify the integral.

### Step 1: Substitution
Let’s make the substitution:

\[
u = \ln(5x)
\]

This implies:

\[
du = \frac{1}{5x} \, dx
\]

This substitution will allow us to simplify the integral. The integral now becomes:

\[
\int \frac{\ln(u)}{5x} \, dx
\]

From the substitution \( u = \ln(5x) \), we know that \( \frac{du}{dx} = \frac{1}{5x} \), so \( dx = 5x \, du \).

### Step 2: Update the Integral
Now the integral becomes:

\[
\int \ln(u) \, du
\]

### Step 3: Integrate
The integral of \(\ln(u)\) is a standard one and can be computed using integration by parts.

Recall that:

\[
\int \ln(u) \, du = u \ln(u) - u + C
\]

### Step 4: Substitute back
Now we substitute back \( u = \ln(5x) \) into the result:

\[
\int \frac{\ln(\ln(5x))}{5x} \, dx = \ln(5x) \ln(\ln(5x)) - \ln(5x) + C
\]

So the solution is:

\[
\boxed{\ln(5x) \ln(\ln(5x)) - \ln(5x) + C}
\]

---

Would you like more details on any part of the solution, or do you have any follow-up questions?

Here are some related questions for further exploration:

1. What is the method of integration by parts and how can it be applied to different types of integrals?
2. Can you show how to integrate a logarithmic function with more complex compositions?
3. How do substitution methods help simplify integrals involving products of functions?
4. What are the conditions under which logarithmic integrals can be solved easily?
5. How can you verify the correctness of an integral after solving?

**Tip:** When working with integrals involving logarithmic and exponential functions, always look for substitution opportunities, as they often simplify complex integrals.

Learn how to integrate the function ln(ln(5x))/5x using substitution and integration by parts. This detailed step-by-step guide covers the process of simplifying complex integrals involving logarithmic functions. Suitable for high school students in Grades 11-12.

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