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!

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.