We are given the integral:

\[
\int \frac{e^{\tan(2x)} + 3}{\cos^2(2x)} \, dx
\]

### Step-by-Step Solution:

1. **Substitution:**
   Let's make the substitution \( u = \tan(2x) \), which means:
   \[
   \frac{du}{dx} = 2 \sec^2(2x) = \frac{2}{\cos^2(2x)}
   \]
   Therefore,
   \[
   dx = \frac{\cos^2(2x)}{2} \, du
   \]

2. **Rewrite the Integral:**
   Using this substitution, we rewrite the integral as:
   \[
   \int \frac{e^{u} + 3}{\cos^2(2x)} \, dx = \frac{1}{2} \int (e^u + 3) \, du
   \]

3. **Integrate:**
   Now, we can integrate:
   \[
   \int e^u \, du = e^u
   \]
   \[
   \int 3 \, du = 3u
   \]
   So, the integral becomes:
   \[
   \frac{1}{2} \left( e^u + 3u \right) + C
   \]

4. **Substitute back \( u = \tan(2x) \):**
   Finally, substitute \( u = \tan(2x) \) back into the expression:
   \[
   \frac{1}{2} \left( e^{\tan(2x)} + 3\tan(2x) \right) + C
   \]

Thus, the solution to the integral is:

\[
\boxed{\frac{1}{2} \left( e^{\tan(2x)} + 3\tan(2x) \right) + C}
\]

Would you like more details on any of these steps or have any further questions?

### Related Questions:
1. How can substitution be used effectively in integration?
2. What is the relationship between trigonometric functions and their derivatives?
3. How do we handle integrals involving exponentials and trigonometric functions together?
4. What are common strategies for integrating functions with \( \tan(x) \) and \( \sec(x) \)?
5. How does the secant function behave in integral calculus?

**Tip:** Always check if a trigonometric substitution simplifies the integrand before attempting more advanced techniques!

Explore the detailed solution to the integral ∫(e^tan(2x)+3)/cos^2(2x) using the substitution method. This solution involves exponential and trigonometric functions, making it ideal for students at the undergraduate calculus level.

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