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Find the integral ∫ (tan(x))^2 dx.

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Question: Find the integral ∫ (tan(x))^2 dx.

Options:

  1. tan(x) - x + C
  2. tan(x) + x + C
  3. tan(x) + x
  4. tan(x) - x

Correct Answer: tan(x) - x + C

Solution: 

Using the identity tan^2(x) = sec^2(x) - 1, we find that ∫ (tan(x))^2 dx = tan(x) - x + C.

Find the integral ∫ (tan(x))^2 dx.

Practice Questions

Q1
Find the integral ∫ (tan(x))^2 dx.
  1. tan(x) - x + C
  2. tan(x) + x + C
  3. tan(x) + x
  4. tan(x) - x

Questions & Step-by-Step Solutions

Find the integral ∫ (tan(x))^2 dx.
  • Step 1: Recall the identity for tangent squared: tan^2(x) = sec^2(x) - 1.
  • Step 2: Substitute this identity into the integral: ∫ (tan(x))^2 dx becomes ∫ (sec^2(x) - 1) dx.
  • Step 3: Split the integral into two parts: ∫ sec^2(x) dx - ∫ 1 dx.
  • Step 4: Calculate the first integral: ∫ sec^2(x) dx = tan(x).
  • Step 5: Calculate the second integral: ∫ 1 dx = x.
  • Step 6: Combine the results: tan(x) - x + C, where C is the constant of integration.
  • Integration of Trigonometric Functions – The question tests the ability to integrate the square of the tangent function using trigonometric identities.
  • Trigonometric Identities – The use of the identity tan^2(x) = sec^2(x) - 1 is crucial for simplifying the integral.
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