To solve the integral of \( 5 \sec^4(x) \tan^8(x) \, dx \), we can approach it by breaking down the expression using trigonometric identities and substitution. Here's a step-by-step process:

### Step 1: Write down the integral
\[
\int 5 \sec^4(x) \tan^8(x) \, dx
\]

### Step 2: Factor out the constant
\[
5 \int \sec^4(x) \tan^8(x) \, dx
\]

### Step 3: Use a trigonometric identity
Recall that:
\[
\sec^2(x) = 1 + \tan^2(x)
\]
Thus, we can rewrite \( \sec^4(x) \) as:
\[
\sec^4(x) = \sec^2(x) \cdot \sec^2(x) = (1 + \tan^2(x)) \cdot \sec^2(x)
\]
Now the integral becomes:
\[
5 \int (1 + \tan^2(x)) \sec^2(x) \tan^8(x) \, dx
\]

### Step 4: Substitution
Let \( u = \tan(x) \), so \( du = \sec^2(x) \, dx \). The integral now becomes:
\[
5 \int (1 + u^2) u^8 \, du
\]
Simplifying the expression inside the integral:
\[
5 \int (u^8 + u^{10}) \, du
\]

### Step 5: Perform the integration
Now, integrate term by term:
\[
5 \left( \frac{u^9}{9} + \frac{u^{11}}{11} \right) + C
\]
Substitute \( u = \tan(x) \) back into the expression:
\[
5 \left( \frac{\tan^9(x)}{9} + \frac{\tan^{11}(x)}{11} \right) + C
\]

### Final Answer:
\[
\boxed{ \frac{5}{9} \tan^9(x) + \frac{5}{11} \tan^{11}(x) + C }
\]

Let me know if you need further clarification or details!

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Here are 5 related questions to expand your understanding:
1. How can substitution help in evaluating trigonometric integrals?
2. What are common identities used in solving integrals involving \( \sec(x) \) and \( \tan(x) \)?
3. What techniques can simplify trigonometric integrals in general?
4. How do you evaluate definite integrals involving trigonometric functions?
5. Can integrals involving powers of other trigonometric functions (e.g., sine, cosine) follow a similar process?

**Tip:** Always look for opportunities to use trigonometric identities when solving integrals—they can greatly simplify the expression!

This guide walks through solving the integral of 5 sec^4(x) tan^8(x) using trigonometric identities and substitution techniques. Learn how to simplify complex integrals involving secant and tangent functions with a step-by-step approach. Suitable for advanced high school and college students.

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