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

### Step 1: Simplify the Integral
First, recall the identity for \(\sec^2(x)\):
\[
\sec^2(x) = 1 + \tan^2(x)
\]
Let:
\[
I = \int \tan^6(2x) \sec^4(2x) \, dx
\]
We can write \(\sec^4(2x)\) as \(\sec^2(2x) \cdot \sec^2(2x)\), and substitute the identity:
\[
I = \int \tan^6(2x) \sec^2(2x) \cdot \sec^2(2x) \, dx
\]
\[
I = \int \tan^6(2x) \sec^2(2x) \cdot (1 + \tan^2(2x)) \, dx
\]
Now expand the integral:
\[
I = \int \tan^6(2x) \sec^2(2x) \, dx + \int \tan^8(2x) \sec^2(2x) \, dx
\]

### Step 2: Use Substitution
For both integrals, let \( u = \tan(2x) \). Then, \( du = 2\sec^2(2x) \, dx \), or equivalently:
\[
\frac{1}{2} du = \sec^2(2x) \, dx
\]

The integrals now become:
\[
I = \frac{1}{2} \int u^6 \, du + \frac{1}{2} \int u^8 \, du
\]

### Step 3: Integrate
Integrate each term separately:
\[
I = \frac{1}{2} \left( \frac{u^7}{7} \right) + \frac{1}{2} \left( \frac{u^9}{9} \right) + C
\]
\[
I = \frac{u^7}{14} + \frac{u^9}{18} + C
\]

### Step 4: Substitute Back \( u = \tan(2x) \)
Finally, substitute back \( u = \tan(2x) \) to get the final answer:
\[
I = \frac{\tan^7(2x)}{14} + \frac{\tan^9(2x)}{18} + C
\]

This is the indefinite integral of \(\tan^6(2x) \sec^4(2x) \, dx\).

Would you like further details or have any questions?

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Here are 5 related questions you might find interesting:
1. How do you integrate \(\int \sec^n(x) \, dx\) for any \(n\)?
2. Can you solve \(\int \tan^4(x) \, dx\)?
3. What are some common trigonometric identities used in integration?
4. How would you approach \(\int \sin^n(x) \cos^m(x) \, dx\)?
5. How can you apply substitution in trigonometric integrals involving multiple angles?

**Tip:** Always double-check your substitutions to ensure that the differential \( du \) corresponds correctly with the original \( dx \) term.

Discover the step-by-step solution to the integral of tan^6(2x) sec^4(2x) dx using trigonometric identities and the substitution method. Understand key concepts in trigonometry and integration, ideal for Grades 11-12 students.

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