To solve the integral \( \int \frac{3}{e^{3x} - e^x} \, dx \), let's go step by step.

### Step 1: Simplify the denominator
The expression in the denominator is \( e^{3x} - e^x \). We can factor this as:

\[
e^{3x} - e^x = e^x(e^{2x} - 1)
\]

### Step 2: Rewrite the integral
The integral now becomes:

\[
\int \frac{3}{e^x(e^{2x} - 1)} \, dx
\]

### Step 3: Substitution
Let \( u = e^x \), then \( du = e^x \, dx \). This simplifies the integral as follows:

\[
\int \frac{3}{u(u^2 - 1)} \, du
\]

Now, we can simplify this using partial fraction decomposition.

### Step 4: Partial fraction decomposition
We decompose the fraction:

\[
\frac{3}{u(u^2 - 1)} = \frac{A}{u} + \frac{B}{u-1} + \frac{C}{u+1}
\]

Multiplying through by \( u(u^2 - 1) = u(u-1)(u+1) \), we get:

\[
3 = A(u-1)(u+1) + B u(u+1) + C u(u-1)
\]

Expanding both sides and equating coefficients gives values for \( A \), \( B \), and \( C \).

### Step 5: Solve for A, B, C
The solution yields:

\[
\frac{3}{u(u^2 - 1)} = \frac{1}{u} - \frac{1}{u-1} - \frac{1}{u+1}
\]

### Step 6: Integrate each term
Now we can integrate term by term:

\[
\int \left( \frac{1}{u} - \frac{1}{u-1} - \frac{1}{u+1} \right) du
\]

This gives:

\[
\ln|u| - \ln|u-1| - \ln|u+1| + C
\]

### Step 7: Back-substitute \( u = e^x \)
Substitute \( u = e^x \) back into the equation:

\[
\ln|e^x| - \ln|e^x - 1| - \ln|e^x + 1| + C
\]

Simplifying further:

\[
x - \ln|e^x - 1| - \ln|e^x + 1| + C
\]

Thus, the solution to the integral is:

\[
\boxed{x - \ln|e^x - 1| - \ln|e^x + 1| + C}
\]

---

Would you like more details on any part of the process?

Here are some related questions:
1. How does partial fraction decomposition work in detail?
2. Why did we choose \( u = e^x \) for substitution?
3. How can we simplify logarithmic expressions involving \( e^x \)?
4. What is the method of integrating rational functions?
5. How can we use substitution more effectively in other integrals?

**Tip:** When integrating fractions involving exponentials, substitution often simplifies the problem drastically.

This page provides a step-by-step solution to the integral of 3/(e^(3x) - e^x). It covers advanced integration techniques like substitution, partial fractions, and logarithmic integration. Perfect for college-level calculus students.

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