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Find the integral ∫ sin(2x) dx.

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

Options:

  1. -cos(2x)/2 + C
  2. cos(2x)/2 + C
  3. -sin(2x)/2 + C
  4. sin(2x)/2 + C

Correct Answer: -cos(2x)/2 + C

Solution: 

The integral of sin(kx) is -1/k * cos(kx). Here, k = 2, so ∫ sin(2x) dx = -cos(2x)/2 + C.

Find the integral ∫ sin(2x) dx.

Practice Questions

Q1
Find the integral ∫ sin(2x) dx.
  1. -cos(2x)/2 + C
  2. cos(2x)/2 + C
  3. -sin(2x)/2 + C
  4. sin(2x)/2 + C

Questions & Step-by-Step Solutions

Find the integral ∫ sin(2x) dx.
  • Step 1: Identify the function to integrate, which is sin(2x).
  • Step 2: Recognize that the integral of sin(kx) is -1/k * cos(kx), where k is a constant.
  • Step 3: In this case, k is 2 because we have sin(2x).
  • Step 4: Substitute k = 2 into the formula: the integral becomes -1/2 * cos(2x).
  • Step 5: Don't forget to add the constant of integration, C, to the result.
  • Step 6: Write the final answer: ∫ sin(2x) dx = -cos(2x)/2 + C.
  • Integration of Trigonometric Functions – Understanding how to integrate functions like sin(kx) using the formula ∫ sin(kx) dx = -1/k * cos(kx).
  • Constant Factor in Integration – Recognizing the role of the constant factor 'k' in the integration process and how it affects the result.
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