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The integral you've provided is:

\[
\int 8 \tan(x) \sec^3(x) \, dx
\]

To solve this, we can use the following method:

### Step 1: Factor out the constant
Since \(8\) is a constant, we can factor it out of the integral:

\[
8 \int \tan(x) \sec^3(x) \, dx
\]

### Step 2: Use a substitution
To evaluate the integral, we can use the substitution \( u = \sec(x) \). The derivative of \( u \) with respect to \( x \) is:

\[
\frac{du}{dx} = \sec(x) \tan(x)
\]

This means:

\[
du = \sec(x) \tan(x) \, dx
\]

### Step 3: Rewrite the integral
Substituting \( u = \sec(x) \) and \( du = \sec(x) \tan(x) \, dx \), the integral becomes:

\[
8 \int u^2 \, du
\]

### Step 4: Integrate
Now, integrate \( u^2 \) with respect to \( u \):

\[
\int u^2 \, du = \frac{u^3}{3}
\]

So the integral becomes:

\[
8 \cdot \frac{u^3}{3} + C = \frac{8}{3} u^3 + C
\]

### Step 5: Substitute back
Substituting back \( u = \sec(x) \), we get:

\[
\frac{8}{3} \sec^3(x) + C
\]

### Final Answer:
\[
\int 8 \tan(x) \sec^3(x) \, dx = \frac{8}{3} \sec^3(x) + C
\]

Would you like more details on any of the steps, or do you have any other questions?

Here are five related questions for further exploration:
1. How do you integrate functions involving \(\sec(x)\) without substitution?
2. What is the relationship between \(\sec(x)\) and \(\cos(x)\)?
3. Can the substitution method be used for other trigonometric integrals?
4. How would the integral change if the integrand was \(8 \sec^3(x)\) only?
5. What happens if you integrate \( \tan^3(x) \sec(x) \, dx \)?

**Tip:** When dealing with trigonometric integrals, substitution is often a powerful technique. Familiarity with trigonometric identities can simplify these processes considerably.

Learn how to solve the integral ∫ 8 tan(x) sec^3(x) dx using substitution and trigonometric identities. This detailed guide covers key calculus concepts suitable for high school and college-level students.

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