Evaluate ∫_0^π/2 cos^2(x) dx.

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Question: Evaluate ∫_0^π/2 cos^2(x) dx.

Options:

Correct Answer: π/4

Solution:

∫_0^π/2 cos^2(x) dx = π/4.

Evaluate ∫_0^π/2 cos^2(x) dx.

Evaluate ∫_0^π/2 cos^2(x) dx.

Step 1: Recognize that we need to evaluate 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. The integral of cos(2x) is (1/2)sin(2x), so evaluate it from 0 to π/2.
Step 7: Evaluate (1/2) * [sin(2x)/2]_0^(π/2) = (1/2) * (sin(π) - sin(0)) = (1/2) * (0 - 0) = 0.
Step 8: Combine the results: π/4 + 0 = π/4.
Step 9: Conclude that ∫_0^(π/2) cos^2(x) dx = π/4.

Integration of Trigonometric Functions – The question tests the ability to integrate the square of the cosine function over a specified interval.
Use of Trigonometric Identities – The solution may require the application of the identity cos^2(x) = (1 + cos(2x))/2 to simplify the integral.
Definite Integrals – The question involves evaluating a definite integral, which requires proper limits of integration.