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.
Questions & Step-by-step Solutions
1 item
Q
Q: Calculate ∫ from 0 to π/2 of sin(x) cos(x) dx.
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.
Steps: 10
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.
