Use the identity (\tan^2(\theta) = \sec^2(\theta) - 1):

Let's use the identity \(\tan^2(\theta) = \sec^2(\theta) - 1\) to solve the integral \(\int (\tan(2x) + \sec(2x))^2 \, dx\).

### 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: Apply the Identity
Substitute \(\tan^2(2x) = \sec^2(2x) - 1\) into the expanded expression:

\[
\tan^2(2x) + 2\tan(2x)\sec(2x) + \sec^2(2x) = (\sec^2(2x) - 1) + 2\tan(2x)\sec(2x) + \sec^2(2x)
\]

Simplify the expression:

\[
\sec^2(2x) - 1 + 2\sec(2x)\tan(2x) + \sec^2(2x) = 2\sec^2(2x) + 2\sec(2x)\tan(2x) - 1
\]

### Step 3: Integrate Term by Term
Now, integrate each term separately.

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

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

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

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

#### Integral of \(-1\)
This is a straightforward linear integral:

\[
\int -1 \, dx = -x
\]

### Step 4: Combine All the Results
Now, combine all the integrals:

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

This is the final answer.

Do you have any questions about these steps, or would you like further details?

### Related Questions:
1. How do you prove the identity \(\tan^2

Learn how to solve the integral \( \int (\tan(2x) + \sec(2x))^2 \, dx \) with a detailed step-by-step solution. This problem covers trigonometric identities, integral calculus techniques, and specific formulas like \( \tan^2(\theta) = \sec^2(\theta) - 1 \). Suitable for advanced high school students.

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