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Calculate ∫ from 0 to π/2 of sin^2(x) dx.
Calculate ∫ from 0 to π/2 of sin^2(x) dx.
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Practice Questions
1 question
Q1
Calculate ∫ from 0 to π/2 of sin^2(x) dx.
π/4
π/2
π/3
π/6
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Using the identity sin^2(x) = (1 - cos(2x))/2, the integral evaluates to π/4.
Questions & Step-by-step Solutions
1 item
Q
Q: Calculate ∫ from 0 to π/2 of sin^2(x) dx.
Solution:
Using the identity sin^2(x) = (1 - cos(2x))/2, the integral evaluates to π/4.
Steps: 8
Show Steps
Step 1: Write down the integral you want to calculate: ∫ from 0 to π/2 of sin^2(x) dx.
Step 2: Use the identity for sin^2(x): sin^2(x) = (1 - cos(2x))/2.
Step 3: Substitute this identity into the integral: ∫ from 0 to π/2 of (1 - cos(2x))/2 dx.
Step 4: Split the integral into two parts: (1/2) * ∫ from 0 to π/2 of 1 dx - (1/2) * ∫ from 0 to π/2 of cos(2x) dx.
Step 5: Calculate the first integral: ∫ from 0 to π/2 of 1 dx = [x] from 0 to π/2 = π/2.
Step 6: Calculate the second integral: ∫ from 0 to π/2 of cos(2x) dx = [sin(2x)/2] from 0 to π/2 = (sin(π) - sin(0))/2 = 0.
Step 7: Combine the results: (1/2) * (π/2) - (1/2) * 0 = π/4.
Step 8: Conclude that the value of the integral ∫ from 0 to π/2 of sin^2(x) dx is π/4.
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