To solve the integral \(\int (\tan(2x) + \sec(2x))^2 \, dx\), we'll follow these steps:

### Step 1: Expand the Square
First, expand the square of the sum:

\[
(\tan(2x) + \sec(2x))^2 = \tan^2(2x) + 2\tan(2x)\sec(2x) + \sec^2(2x)
\]

### Step 2: Integrate Term by Term
Now, integrate each term separately.

#### Integral of \(\tan^2(2x)\)
Use the identity \(\tan^2(\theta) = \sec^2(\theta) - 1\):

\[
\int \tan^2(2x) \, dx = \int (\sec^2(2x) - 1) \, dx
\]

Integrating:

\[
\int (\sec^2(2x) - 1) \, dx = \frac{1}{2}\tan(2x) - x
\]

#### Integral of \(2\tan(2x)\sec(2x)\)
Recognize that \(2\tan(2x)\sec(2x)\) is the derivative of \(\sec(2x)\):

\[
\int 2\tan(2x)\sec(2x) \, dx = \sec(2x)
\]

#### Integral of \(\sec^2(2x)\)
The integral of \(\sec^2(\theta)\) is \(\tan(\theta)\):

\[
\int \sec^2(2x) \, dx = \frac{1}{2} \tan(2x)
\]

### Step 3: Combine All the Results
Now, combine all the integrals:

\[
\int (\tan(2x) + \sec(2x))^2 \, dx = \frac{1}{2}\tan(2x) - x + \sec(2x) + \frac{1}{2}\tan(2x) + C
\]

Simplify:

\[
\int (\tan(2x) + \sec(2x))^2 \, dx = \tan(2x) + \sec(2x) - x + C
\]

This is the final answer.

Would you like more details on any of the steps, or do you have any questions?

### Related Questions:
1. How do you derive the integral of \(\sec(x)\)?
2. What are other methods for integrating \(\tan^2(x)\)?
3. How can substitution be applied to solve integrals involving trigonometric functions?
4. What is the integral of \(\sec^3(x)\)?
5. How do identities simplify the integration of trigonometric functions?

### Tip:
When integrating trigonometric functions, it's often helpful to remember key identities, such as \(\tan^2(x) = \sec^2(x) - 1\), to simplify the process.

Explore how to integrate the square of the sum of tan(2x) and sec(2x) with a detailed step-by-step solution. This problem involves integral calculus and trigonometric functions, covering integration by parts and key trigonometric identities. Suitable for advanced college-level students and learners.

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