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Calculate ∫ from 0 to π/2 of sin(x) cos(x) dx.

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Question: Calculate ∫ from 0 to π/2 of sin(x) cos(x) dx.

Options:

  1. 1/2
  2. 1
  3. π/4
  4. π/2

Correct Answer: 1/2

Solution: 

Using the identity sin(2x) = 2sin(x)cos(x), the integral becomes 1/2 ∫ from 0 to π/2 of sin(2x) dx = 1/2 [-1/2 cos(2x)] from 0 to π/2 = 1/2 [0 - (-1/2)] = 1/4.

Calculate ∫ from 0 to π/2 of sin(x) cos(x) dx.

Practice Questions

Q1
Calculate ∫ from 0 to π/2 of sin(x) cos(x) dx.
  1. 1/2
  2. 1
  3. π/4
  4. π/2

Questions & Step-by-Step Solutions

Calculate ∫ from 0 to π/2 of sin(x) cos(x) dx.
  • Step 1: Start with the integral you want to calculate: ∫ from 0 to π/2 of sin(x) cos(x) dx.
  • Step 2: Use the trigonometric identity sin(2x) = 2sin(x)cos(x) to rewrite the integral. This means sin(x)cos(x) = (1/2)sin(2x).
  • Step 3: Substitute this into the integral: ∫ from 0 to π/2 of sin(x) cos(x) dx becomes (1/2) ∫ from 0 to π/2 of sin(2x) dx.
  • Step 4: Now, calculate the integral ∫ sin(2x) dx. The antiderivative of sin(2x) is -1/2 cos(2x).
  • Step 5: Apply the limits of integration from 0 to π/2: (1/2) [-1/2 cos(2x)] from 0 to π/2.
  • Step 6: Evaluate at the upper limit π/2: -1/2 cos(2(π/2)) = -1/2 cos(π) = -1/2 * (-1) = 1/2.
  • Step 7: Evaluate at the lower limit 0: -1/2 cos(2(0)) = -1/2 cos(0) = -1/2 * 1 = -1/2.
  • Step 8: Now, subtract the lower limit result from the upper limit result: (1/2) - (-1/2) = (1/2) + (1/2) = 1.
  • Step 9: Multiply by (1/2) from Step 3: (1/2) * 1 = 1/2.
  • Step 10: Therefore, the final answer is 1/4.
  • Trigonometric Identities – Understanding and applying the identity sin(2x) = 2sin(x)cos(x) to simplify the integral.
  • Definite Integrals – Calculating the value of a definite integral by evaluating the antiderivative at the bounds.
  • Integration Techniques – Using substitution and integration of trigonometric functions to solve the integral.
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