Evaluate the integral.

Integral from nothing to nothing 8 tangent x secant Superscript 5 Baseline x font size decreased by 3 dx∫8tanxsec5x dx

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Part 1

Integral from nothing to nothing 8 tangent x secant Superscript 5 Baseline x font size decreased by 3 dx∫8tanxsec5x dxequals=enter your response here

We are tasked with evaluating the integral:

\[
\int 8 \tan(x) \sec^5(x) \, dx
\]

### Step-by-Step Solution:

1. **Identify the Structure of the Integral**: The given integral involves the product of \(\tan(x)\) and \(\sec^5(x)\). Since both functions are related through trigonometric identities, we can simplify the integration by expressing it in a more integrable form.

2. **Rewrite the Integral**:
   We can split the powers of \(\sec(x)\) and use the identity \(\sec^2(x) = 1 + \tan^2(x)\). Let's express the integral as:
   \[
   \int 8 \tan(x) \sec^5(x) \, dx = 8 \int \tan(x) \sec^4(x) \cdot \sec(x) \, dx
   \]

3. **Substitute for Trigonometric Function**:
   We now use a substitution to simplify the integral. Let:
   \[
   u = \sec(x), \quad \text{then} \, du = \sec(x) \tan(x) \, dx
   \]
   This substitution transforms the integral as follows:
   \[
   \int 8 \sec(x) \sec^4(x) \tan(x) \, dx = 8 \int u^4 \, du
   \]

4. **Integrate**:
   The integral of \(u^4\) is straightforward:
   \[
   8 \int u^4 \, du = 8 \cdot \frac{u^5}{5} = \frac{8}{5} u^5
   \]

5. **Substitute Back**:
   Now, substitute \(u = \sec(x)\) back into the expression:
   \[
   \frac{8}{5} \sec^5(x) + C
   \]
   where \(C\) is the constant of integration.

### Final Answer:
\[
\int 8 \tan(x) \sec^5(x) \, dx = \frac{8}{5} \sec^5(x) + C
\]

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Would you like further details on any step or have additional questions?

Here are five related questions you might explore:
1. What is the derivative of \(\sec(x)\)?
2. How can we apply trigonometric identities to simplify complex integrals?
3. What is the integral of \(\sec^n(x)\) for other values of \(n\)?
4. How is substitution typically used to solve integrals involving trigonometric functions?
5. What are some common applications of integrals involving trigonometric functions?

**Tip:** When integrating trigonometric functions, always check if substitution or known identities can simplify the process.

Evaluate the integral: ∫ 8 tan(x) sec^5(x) dx

This problem demonstrates how to evaluate the integral ∫ 8 tan(x) sec^5(x) dx using trigonometric substitution and identities. Learn the step-by-step method to simplify and integrate functions involving secant and tangent, suitable for undergraduate calculus students.

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