Calculate ∫ from 0 to π/2 of sin^2(x) dx.

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Question: Calculate ∫ from 0 to π/2 of sin^2(x) dx.

Options:

Correct Answer: π/4

Solution:

Using the identity sin^2(x) = (1 - cos(2x))/2, the integral evaluates to π/4.

Calculate ∫ from 0 to π/2 of sin^2(x) dx.

Calculate ∫ from 0 to π/2 of sin^2(x) dx.

Correct Answer: π/4

Step 1: Write down the integral you want to calculate: ∫ from 0 to π/2 of 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: ∫ from 0 to π/2 of (1 - cos(2x))/2 dx.
Step 4: Split the integral into two parts: (1/2) * ∫ from 0 to π/2 of 1 dx - (1/2) * ∫ from 0 to π/2 of cos(2x) dx.
Step 5: Calculate the first integral: ∫ from 0 to π/2 of 1 dx = [x] from 0 to π/2 = π/2.
Step 6: Calculate the second integral: ∫ from 0 to π/2 of cos(2x) dx = [sin(2x)/2] from 0 to π/2 = (sin(π) - sin(0))/2 = 0.
Step 7: Combine the results: (1/2) * (π/2) - (1/2) * 0 = π/4.
Step 8: Conclude that the value of the integral ∫ from 0 to π/2 of sin^2(x) dx is π/4.

Trigonometric Identities – Understanding and applying the identity sin^2(x) = (1 - cos(2x))/2 to simplify the integral.
Definite Integrals – Calculating the area under the curve of a function over a specified interval.