Evaluate the integral. (Remember the constant of integration. 8 tan^3(x) sec(x) dx



We are tasked with evaluating the integral:

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

### Step 1: Rewrite \( \tan^3(x) \)
We can express \( \tan^3(x) \) as \( \tan(x) \cdot \tan^2(x) \), and we know from trigonometric identities that:

\[
\tan^2(x) = \sec^2(x) - 1
\]

Thus, we rewrite the integrand as:

\[
8 \int \tan(x) \cdot (\sec^2(x) - 1) \sec(x) \, dx
\]

### Step 2: Expand the integrand
Distribute the \( \tan(x) \) term:

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

Now we have two integrals:

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

### Step 3: Evaluate each integral

#### Integral 1: \( \int \tan(x) \sec^3(x) \, dx \)

We use substitution here. Recall that \( \frac{d}{dx}(\sec(x)) = \sec(x) \tan(x) \), so we let:

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

This gives:

\[
\int \tan(x) \sec^3(x) \, dx = \int u^2 \, du = \frac{u^3}{3} = \frac{\sec^3(x)}{3}
\]

#### Integral 2: \( \int \tan(x) \sec(x) \, dx \)

This is simpler, as \( \frac{d}{dx}(\sec(x)) = \sec(x) \tan(x) \), so:

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

### Step 4: Combine the results

Now we substitute the evaluated integrals back into the expression:

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

Simplifying:

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

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

Would you like further clarification on any step or more details?

### Related Questions:
1. How would this integral change if the power of \( \tan(x) \) were different?
2. What other trigonometric identities can simplify integrals involving \( \sec(x) \) and \( \tan(x) \)?
3. How do we apply integration by parts to trigonometric functions?
4. What is the process for solving integrals involving \( \sin(x) \) and \( \cos(x) \) with powers?
5. How can we use substitution in other trigonometric integrals?

### Tip:
When working with integrals of products involving trigonometric functions, identify identities or substitutions that can simplify the expressions first.

Evaluate the integral. (Remember the constant of integration. 8 tan^3(x) sec(x) dx

Learn how to evaluate the integral of 8 tan^3(x) sec(x) dx with a step-by-step solution. This calculus problem covers key trigonometric identities and integration techniques, suitable for advanced high school or undergraduate students.

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