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To evaluate the integral \( \int -4 \tan^4(x) \, dx \), we can approach it by simplifying and breaking down the power of the tangent function. Here's how we proceed step by step:

### Step 1: Use trigonometric identities to simplify the integrand
We can express \( \tan^4(x) \) in terms of lower powers of trigonometric functions. Recall the identity:
\[
\tan^2(x) = \sec^2(x) - 1
\]
Thus, \( \tan^4(x) \) can be written as:
\[
\tan^4(x) = (\tan^2(x))^2 = (\sec^2(x) - 1)^2
\]
Now expand this expression:
\[
\tan^4(x) = \sec^4(x) - 2\sec^2(x) + 1
\]

### Step 2: Substitute this into the original integral
The integral becomes:
\[
\int -4 \tan^4(x) \, dx = \int -4 (\sec^4(x) - 2 \sec^2(x) + 1) \, dx
\]
Distribute the \(-4\):
\[
= \int (-4 \sec^4(x) + 8 \sec^2(x) - 4) \, dx
\]

### Step 3: Break the integral into parts
Now, we handle each part separately:
\[
\int (-4 \sec^4(x) + 8 \sec^2(x) - 4) \, dx = -4 \int \sec^4(x) \, dx + 8 \int \sec^2(x) \, dx - 4 \int 1 \, dx
\]

### Step 4: Evaluate each integral
1. **Integral of \( \sec^4(x) \)**:
   Use the reduction formula for \( \sec^n(x) \):
   \[
   \int \sec^4(x) \, dx = \frac{\tan(x) \sec^2(x)}{3} + \frac{2}{3} \int \sec^2(x) \, dx
   \]
   We will need this result shortly.

2. **Integral of \( \sec^2(x) \)**:
   The integral of \( \sec^2(x) \) is straightforward:
   \[
   \int \sec^2(x) \, dx = \tan(x)
   \]

3. **Integral of \( 1 \)**:
   This is simply:
   \[
   \int 1 \, dx = x
   \]

### Step 5: Put everything together
Using the evaluated parts, we combine the results:
\[
-4 \left( \frac{\tan(x) \sec^2(x)}{3} + \frac{2}{3} \tan(x) \right) + 8 \tan(x) - 4x
\]

Thus, the solution to the integral is:
\[
- \frac{4}{3} \tan(x) \sec^2(x) - \frac{8}{3} \tan(x) + 8 \tan(x) - 4x + C
\]
where \( C \) is the constant of integration.

Let me know if you would like further details or explanations!

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Here are 5 questions for you:
1. How would you evaluate the integral of \( \sec^n(x) \) for \( n \geq 3 \)?
2. What is the reduction formula for \( \sec^n(x) \) in general?
3. Can you explain how the identity \( \tan^2(x) = \sec^2(x) - 1 \) is derived?
4. How does the method of integration by parts help in solving trigonometric integrals?
5. What other techniques could simplify higher-order powers of trigonometric functions in integrals?

**Tip**: When faced with higher powers of trigonometric functions in integrals, always try to reduce their powers using identities to simpler forms.

Step-by-step solution to evaluate the integral of -4 tan^4(x) using trigonometric identities and reduction formulas. This problem is ideal for students in advanced high school calculus or university-level math courses.

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