Find the value of ∫_0^π/2 cos^2(x) dx.

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Question: Find the value of ∫_0^π/2 cos^2(x) dx.

Options:

Correct Answer: π/4

Solution:

The integral evaluates to [x/2 + sin(2x)/4] from 0 to π/2 = π/4.

Find the value of ∫_0^π/2 cos^2(x) dx.

Find the value of ∫_0^π/2 cos^2(x) dx.

Step 1: Identify the integral we need to solve: ∫_0^(π/2) cos^2(x) dx.
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] from 0 to 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] from 0 to π/2 = (1/2) * (sin(π) - sin(0)) = 0.
Step 7: Add the results of both integrals: π/4 + 0 = π/4.
Step 8: Conclude that the value of the integral ∫_0^(π/2) cos^2(x) dx is 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 involve using the identity cos^2(x) = (1 + cos(2x))/2 to simplify the integral.
Definite Integrals – The question requires evaluating a definite integral, which involves calculating the antiderivative and applying the limits.