To integrate the function \(\frac{3}{5x - 2}\), we can use a standard integration technique. This is a basic integral that involves a linear function in the denominator.

### Given Integral:
\[
\int \frac{3}{5x - 2} \, dx
\]

### Step-by-Step Solution:

1. **Identify the substitution**:  
   Let's set \(u = 5x - 2\). Then, \(du = 5 \, dx\), or equivalently, \(dx = \frac{du}{5}\).

2. **Rewrite the integral in terms of \(u\)**:  
   Substitute \(u = 5x - 2\) and \(dx = \frac{du}{5}\) into the integral:

   \[
   \int \frac{3}{u} \cdot \frac{du}{5} = \frac{3}{5} \int \frac{1}{u} \, du
   \]

3. **Integrate**:  
   The integral of \(\frac{1}{u}\) with respect to \(u\) is \(\ln |u|\):

   \[
   \frac{3}{5} \ln |u| + C
   \]

4. **Substitute back \(u = 5x - 2\)**:  
   Replace \(u\) with the original expression \(5x - 2\):

   \[
   \frac{3}{5} \ln |5x - 2| + C
   \]

### Final Answer:
\[
\int \frac{3}{5x - 2} \, dx = \frac{3}{5} \ln |5x - 2| + C
\]

where \(C\) is the constant of integration.

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

### Related Questions:

1. How would you integrate \(\frac{1}{5x - 2}\) instead?
2. How do you handle integrals with more complex linear functions in the denominator?
3. What substitution method would be used for \(\int \frac{5}{x^2 + 4x + 4} \, dx\)?
4. How can logarithmic integration be applied to integrals with different exponents?
5. What are the properties of logarithms that help in simplifying integration results?

### Tip:
Always check for possible substitutions that simplify the integrand before proceeding with standard integration techniques!

Learn how to integrate the function 3/(5x - 2) using the substitution method in calculus. This detailed solution covers the steps involved in substituting u = 5x - 2 and integrating with respect to u. Suitable for college-level students and anyone studying calculus.

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