To solve the integral \(\int (\ln(5x))^2 \, dx\), we can use the method of integration by parts. Here's how it can be done:

Let:
\[
u = (\ln(5x))^2 \quad \text{and} \quad dv = dx
\]

Now, we'll differentiate \(u\) and integrate \(dv\):

\[
du = 2 \ln(5x) \cdot \frac{1}{x} \, dx = \frac{2 \ln(5x)}{x} \, dx
\]

\[
v = x
\]

Now, apply the integration by parts formula:
\[
\int u \, dv = uv - \int v \, du
\]

Substituting the values we have:
\[
\int (\ln(5x))^2 \, dx = x(\ln(5x))^2 - \int x \cdot \frac{2 \ln(5x)}{x} \, dx
\]

Simplify the integral:
\[
\int (\ln(5x))^2 \, dx = x(\ln(5x))^2 - 2 \int \ln(5x) \, dx
\]

Next, we need to solve \(\int \ln(5x) \, dx\). Again, we'll use integration by parts:

Let:
\[
u = \ln(5x) \quad \text{and} \quad dv = dx
\]

Differentiate \(u\) and integrate \(dv\):
\[
du = \frac{1}{x} \, dx \quad \text{and} \quad v = x
\]

Now, apply the integration by parts formula:
\[
\int \ln(5x) \, dx = x\ln(5x) - \int x \cdot \frac{1}{x} \, dx
\]

Simplify:
\[
\int \ln(5x) \, dx = x\ln(5x) - \int 1 \, dx
\]
\[
\int \ln(5x) \, dx = x\ln(5x) - x + C
\]

Now, substitute this back into the original expression:
\[
\int (\ln(5x))^2 \, dx = x(\ln(5x))^2 - 2\left(x\ln(5x) - x\right) + C
\]

Finally, simplify the expression:
\[
\int (\ln(5x))^2 \, dx = x(\ln(5x))^2 - 2x\ln(5x) + 2x + C
\]

So, the final answer is:
\[
\boxed{\int (\ln(5x))^2 \, dx = x(\ln(5x))^2 - 2x\ln(5x) + 2x + C}
\]

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

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Here are 5 related questions you might find interesting:

1. How would the integral change if the expression was \(\int (\ln(ax))^n \, dx\)?
2. Can you derive the integral \(\int \ln(x)^2 \, dx\) using a similar method?
3. What if the integrand was \(\int \ln(5x) \, dx\)? How would you approach it?
4. How do you solve \(\int x(\ln(5x))^2 \, dx\)?
5. How does the substitution method compare with integration by parts in solving logarithmic integrals?

**Tip:** When using integration by parts, choosing the correct \(u\) and \(dv\) can simplify the process significantly, especially for logarithmic and polynomial functions.

Explore the detailed step-by-step solution to evaluate the integral of (ln(5x))^2 using integration by parts. This problem involves integrating a logarithmic function squared, showcasing advanced techniques in calculus.

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