Evaluate ∫_0^π/2 sin^2(x) dx.

Evaluate ∫_0^π/2 sin^2(x) dx.

Step 1: Start with the integral you want to evaluate: ∫_0^(π/2) 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: ∫_0^(π/2) sin^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 = [ (1/2) sin(2x) ]_0^(π/2) = (1/2)(sin(π) - sin(0)) = 0.
Step 8: Combine the results: = (1/2) [ (π/2) - 0 ] = (1/2)(π/2) = π/4.

Trigonometric Identities – Understanding and applying the identity sin^2(x) = (1 - cos(2x))/2 to simplify integrals.
Definite Integrals – Evaluating integrals over a specified interval, particularly from 0 to π/2 in this case.