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Calculate ∫_0^π/2 cos^2(x) dx.
Calculate ∫_0^π/2 cos^2(x) dx.
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Practice Questions
1 question
Q1
Calculate ∫_0^π/2 cos^2(x) dx.
π/4
π/2
1
0
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∫_0^π/2 cos^2(x) dx = π/4.
Questions & Step-by-step Solutions
1 item
Q
Q: Calculate ∫_0^π/2 cos^2(x) dx.
Solution:
∫_0^π/2 cos^2(x) dx = π/4.
Steps: 8
Show Steps
Step 1: Recognize that we need to calculate the integral of cos^2(x) from 0 to π/2.
Step 2: Use the trigonometric identity: cos^2(x) = (1 + cos(2x)) / 2.
Step 3: Rewrite the integral using the identity: ∫_0^(π/2) cos^2(x) dx = ∫_0^(π/2) (1 + cos(2x)) / 2 dx.
Step 4: Split the integral into two parts: ∫_0^(π/2) (1/2) dx + ∫_0^(π/2) (cos(2x)/2) dx.
Step 5: Calculate the first integral: ∫_0^(π/2) (1/2) dx = (1/2) * [x]_0^(π/2) = (1/2) * (π/2 - 0) = π/4.
Step 6: Calculate the second integral: ∫_0^(π/2) (cos(2x)/2) dx = (1/2) * [sin(2x)/2]_0^(π/2) = (1/2) * (0 - 0) = 0.
Step 7: Add the results of the two integrals: π/4 + 0 = π/4.
Step 8: Conclude that ∫_0^(π/2) cos^2(x) dx = π/4.
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