To solve the integral of \( \tan^3(4x) \, dx \), we can use trigonometric identities and techniques. Here's the step-by-step solution:

### Step 1: Rewrite \( \tan^3(4x) \)
We can split \( \tan^3(4x) \) as:
\[
\tan^3(4x) = \tan(4x) \cdot \tan^2(4x)
\]
Using the identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \), we get:
\[
\tan^3(4x) = \tan(4x) \cdot (\sec^2(4x) - 1)
\]
Thus, the integral becomes:
\[
\int \tan^3(4x) \, dx = \int \tan(4x) \cdot (\sec^2(4x) - 1) \, dx
\]

### Step 2: Split the integral
Now, we split the integral into two parts:
\[
\int \tan^3(4x) \, dx = \int \tan(4x) \sec^2(4x) \, dx - \int \tan(4x) \, dx
\]

### Step 3: Solve the first integral \( \int \tan(4x) \sec^2(4x) \, dx \)
For the first integral, we can use a substitution. Let \( u = \tan(4x) \), so \( du = 4\sec^2(4x) \, dx \), or \( \frac{du}{4} = \sec^2(4x) \, dx \).

Thus, the integral becomes:
\[
\int \tan(4x) \sec^2(4x) \, dx = \frac{1}{4} \int u \, du = \frac{1}{4} \cdot \frac{u^2}{2} = \frac{\tan^2(4x)}{8}
\]

### Step 4: Solve the second integral \( \int \tan(4x) \, dx \)
The integral of \( \tan(4x) \) is:
\[
\int \tan(4x) \, dx = -\frac{1}{4} \ln|\cos(4x)|
\]

### Step 5: Combine the results
Now, combining both integrals, we get:
\[
\int \tan^3(4x) \, dx = \frac{\tan^2(4x)}{8} - \frac{1}{4} \ln|\cos(4x)| + C
\]

Thus, the final answer is:
\[
\int \tan^3(4x) \, dx = \frac{\tan^2(4x)}{8} - \frac{1}{4} \ln|\cos(4x)| + C
\]

Would you like further details or have any questions about the steps?

Here are five related questions to expand your knowledge:
1. How do you derive the identity \( \tan^2(x) = \sec^2(x) - 1 \)?
2. How do you find the integral of \( \tan^n(x) \) for higher powers of \( n \)?
3. What is the integral of \( \sec^2(4x) \)?
4. How does substitution work when integrating trigonometric functions?
5. Can you explain the importance of the constant \( C \) in indefinite integrals?

**Tip:** When integrating powers of trigonometric functions, always look for useful identities to simplify the integrand before applying integration techniques.

Explore the step-by-step solution to integrate tan^3(4x) using trigonometric identities and substitution. This problem involves concepts of trigonometry and integration, suitable for students in Grades 11-12 or early college. Learn key formulas such as tan^2(θ) = sec^2(θ) - 1 and how to apply substitution in integrals.

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