Math Problem Statement

Int of (\tan^2(x) \sec(x))

Solution

To solve the integral tan2(x)sec(x)dx\int \tan^2(x) \sec(x) \, dx, we'll approach it step by step.

Step 1: Express tan2(x)\tan^2(x) in terms of sec(x)\sec(x)

We know the identity: tan2(x)=sec2(x)1\tan^2(x) = \sec^2(x) - 1 Substituting this into the integral, we have: tan2(x)sec(x)dx=(sec2(x)1)sec(x)dx\int \tan^2(x) \sec(x) \, dx = \int (\sec^2(x) - 1) \sec(x) \, dx

Step 2: Distribute the sec(x)\sec(x)

This gives us: (sec2(x)sec(x)sec(x))dx\int (\sec^2(x) \sec(x) - \sec(x)) \, dx Simplify this to: (sec3(x)sec(x))dx\int (\sec^3(x) - \sec(x)) \, dx

Step 3: Split the integral

Now, we can split the integral into two separate integrals: sec3(x)dxsec(x)dx\int \sec^3(x) \, dx - \int \sec(x) \, dx

Step 4: Evaluate each integral

  1. Integral of sec(x)\sec(x): sec(x)dx=lnsec(x)+tan(x)+C1\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C_1

  2. Integral of sec3(x)\sec^3(x): To solve sec3(x)dx\int \sec^3(x) \, dx, we use the reduction formula for secn(x)\sec^n(x): secn(x)dx=secn2(x)tan(x)n1+n2n1secn2(x)dx\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=3n = 3, we have: sec3(x)dx=sec(x)tan(x)2+12sec(x)dx\int \sec^3(x) \, dx = \frac{\sec(x) \tan(x)}{2} + \frac{1}{2} \int \sec(x) \, dx Substituting the integral of sec(x)\sec(x): sec3(x)dx=sec(x)tan(x)2+12lnsec(x)+tan(x)+C2\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: tan2(x)sec(x)dx=sec(x)tan(x)2+12lnsec(x)+tan(x)lnsec(x)+tan(x)+C\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: tan2(x)sec(x)dx=sec(x)tan(x)212lnsec(x)+tan(x)+C\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?


Here are 8 related questions:

  1. How do you derive the reduction formula for secn(x)\sec^n(x)?
  2. What is the integral of sec2(x)\sec^2(x)?
  3. How would you integrate tan2(x)\tan^2(x)?
  4. What is the identity for tan2(x)\tan^2(x) in terms of sec(x)\sec(x)?
  5. Can you explain the steps to integrate sec3(x)\sec^3(x) more thoroughly?
  6. How does the integral of sec(x)\sec(x) relate to the hyperbolic functions?
  7. What is the geometric interpretation of the integral of sec(x)\sec(x)?
  8. How would the integration process change if the integrand were tan3(x)sec(x)\tan^3(x) \sec(x)?

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

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Integration
Trigonometric Identities

Formulas

Reduction formula for \(\sec^n(x)\)
Integral of \(\sec(x)\)

Theorems

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Suitable Grade Level

Advanced High School