To solve the integral \(\int \tan^2(x) \sec(x) \, dx\), we'll approach it step by step.

### Step 1: Express \(\tan^2(x)\) in terms of \(\sec(x)\)
We know the identity:
\[
\tan^2(x) = \sec^2(x) - 1
\]
Substituting this into the integral, we have:
\[
\int \tan^2(x) \sec(x) \, dx = \int (\sec^2(x) - 1) \sec(x) \, dx
\]

### Step 2: Distribute the \(\sec(x)\)
This gives us:
\[
\int (\sec^2(x) \sec(x) - \sec(x)) \, dx
\]
Simplify this to:
\[
\int (\sec^3(x) - \sec(x)) \, dx
\]

### Step 3: Split the integral
Now, we can split the integral into two separate integrals:
\[
\int \sec^3(x) \, dx - \int \sec(x) \, dx
\]

### Step 4: Evaluate each integral
1. **Integral of \(\sec(x)\)**:
   \[
   \int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C_1
   \]

2. **Integral of \(\sec^3(x)\)**:
   To solve \(\int \sec^3(x) \, dx\), we use the reduction formula for \(\sec^n(x)\):
   \[
   \int \sec^n(x) \, dx = \frac{\sec^{n-2}(x) \tan(x)}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2}(x) \, dx
   \]
   For \(n = 3\), we have:
   \[
   \int \sec^3(x) \, dx = \frac{\sec(x) \tan(x)}{2} + \frac{1}{2} \int \sec(x) \, dx
   \]
   Substituting the integral of \(\sec(x)\):
   \[
   \int \sec^3(x) \, dx = \frac{\sec(x) \tan(x)}{2} + \frac{1}{2} \ln|\sec(x) + \tan(x)| + C_2
   \]

### Step 5: Combine the results
Finally, substituting back into our original expression:
\[
\int \tan^2(x) \sec(x) \, dx = \frac{\sec(x) \tan(x)}{2} + \frac{1}{2} \ln|\sec(x) + \tan(x)| - \ln|\sec(x) + \tan(x)| + C
\]
Simplify:
\[
\int \tan^2(x) \sec(x) \, dx = \frac{\sec(x) \tan(x)}{2} - \frac{1}{2} \ln|\sec(x) + \tan(x)| + C
\]

This is the final answer.

Would you like any more details or have any questions?

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Here are 8 related questions:
1. How do you derive the reduction formula for \(\sec^n(x)\)?
2. What is the integral of \(\sec^2(x)\)?
3. How would you integrate \(\tan^2(x)\)?
4. What is the identity for \(\tan^2(x)\) in terms of \(\sec(x)\)?
5. Can you explain the steps to integrate \(\sec^3(x)\) more thoroughly?
6. How does the integral of \(\sec(x)\) relate to the hyperbolic functions?
7. What is the geometric interpretation of the integral of \(\sec(x)\)?
8. How would the integration process change if the integrand were \(\tan^3(x) \sec(x)\)?

**Tip:** When dealing with trigonometric integrals, consider using trigonometric identities to simplify the integrand before integration.

Explore how to integrate tan^2(x) sec(x) with a detailed step-by-step solution. Learn the reduction formula for \(\sec^n(x)\), integral of \(\sec(x)\), and trigonometric identities. Suitable for advanced high school students and anyone interested in calculus and trigonometry.

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