Evaluate the integral ∫_0^π/2 cos^2(x) dx.

Evaluate the integral ∫_0^π/2 cos^2(x) dx.

Step 1: Recall the integral we want to evaluate: ∫_0^(π/2) cos^2(x) dx.
Step 2: Use the trigonometric identity: cos^2(x) = (1 + cos(2x)) / 2.
Step 3: Substitute this identity into the integral: ∫_0^(π/2) cos^2(x) dx = ∫_0^(π/2) (1 + cos(2x)) / 2 dx.
Step 4: Factor out the 1/2 from the integral: = 1/2 ∫_0^(π/2) (1 + cos(2x)) dx.
Step 5: Split the integral into two parts: = 1/2 (∫_0^(π/2) 1 dx + ∫_0^(π/2) cos(2x) dx).
Step 6: Evaluate the first integral: ∫_0^(π/2) 1 dx = [x]_0^(π/2) = π/2.
Step 7: Evaluate the second integral: ∫_0^(π/2) cos(2x) dx = [sin(2x)/2]_0^(π/2) = (sin(π) - sin(0))/2 = 0.
Step 8: Combine the results: = 1/2 (π/2 + 0) = π/4.

Trigonometric Integration – The question tests the ability to integrate trigonometric functions, specifically the square of the cosine function over a specified interval.
Use of Trigonometric Identities – The solution may require the use of trigonometric identities, such as the identity for cos^2(x) = (1 + cos(2x))/2, to simplify the integral.
Definite Integrals – The question involves evaluating a definite integral, which requires proper application of limits and understanding of the area under the curve.