To evaluate the integral of the function \( \int \tan^6(2x) \sec^4(2x) \, dx \), we can use substitution and trigonometric identities.

### Step 1: Substitution
Let \( u = 2x \). Then, \( du = 2dx \), or \( dx = \frac{du}{2} \).

The integral becomes:
\[
\frac{1}{2} \int \tan^6(u) \sec^4(u) \, du
\]

### Step 2: Simplify the integrand
We can express \( \tan^6(u) \) as \( (\tan^2(u))^3 \) and use the identity \( \sec^2(u) = 1 + \tan^2(u) \). Also, note that \( \sec^4(u) = (\sec^2(u))^2 \).

Thus:
\[
\frac{1}{2} \int \tan^6(u) \sec^4(u) \, du = \frac{1}{2} \int (\tan^2(u))^3 (\sec^2(u))^2 \, du
\]

### Step 3: Express everything in terms of \( \tan(u) \)
Let \( v = \tan(u) \), so \( dv = \sec^2(u) du \).

The integral becomes:
\[
\frac{1}{2} \int v^6 \cdot \sec^2(u) \cdot \sec^2(u) \, du = \frac{1}{2} \int v^6 \cdot dv
\]

### Step 4: Integrate
Now, integrate:
\[
\frac{1}{2} \int v^6 \, dv = \frac{1}{2} \cdot \frac{v^7}{7} = \frac{v^7}{14}
\]

### Step 5: Back-substitute
Substituting back \( v = \tan(u) \) and \( u = 2x \):
\[
\frac{v^7}{14} = \frac{(\tan(2x))^7}{14}
\]

### Final Answer
The integral is:
\[
\int \tan^6(2x) \sec^4(2x) \, dx = \frac{\tan^7(2x)}{14} + C
\]
where \( C \) is the constant of integration.

Would you like further details or have any questions?

Here are five related questions you might find interesting:
1. How would you evaluate \( \int \sec^4(2x) \, dx \)?
2. What substitution would you use for \( \int \tan^3(x) \sec(x) \, dx \)?
3. How do trigonometric identities help simplify integrals?
4. Can you derive the integral of \( \int \tan^2(x) \sec^2(x) \, dx \) directly?
5. What is the integral of \( \int \sec^2(3x) \, dx \)?

**Tip:** When integrating trigonometric functions, substitution and identities often simplify the process significantly.

Learn how to evaluate the integral of tan^6(2x) sec^4(2x) dx using substitution and trigonometric identities with a detailed step-by-step solution. This problem involves advanced integration techniques suitable for high school and college students.

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