Question

Evaluate the definite integral: \( \int_{0}^{\pi/2} \sin^2(x) \, dx \).

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{2} \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

Whenever you encounter powers of trigonometric functions, power-reduction identities are usually the most efficient path to simplification.

Answer 1

The Correct Option is A

Concept: To integrate $\sin^2(x)$, we use the trigonometric power-reduction identity: \[ \sin^2(x) = \frac{1 - \cos(2x)}{2} \]

Step 1: Substitute the identity into the integral. \[ \int_{0}^{\pi/2} \sin^2(x) \, dx = \int_{0}^{\pi/2} \frac{1 - \cos(2x)}{2} \, dx \] \[ = \frac{1}{2} \int_{0}^{\pi/2} (1 - \cos(2x)) \, dx \]

Step 2: Perform the integration. \[ = \frac{1}{2} \left[ x - \frac{\sin(2x)}{2} \right]_{0}^{\pi/2} \]

Step 3: Evaluate at the bounds. At upper bound \( x = \pi/2 \): \( \frac{\pi}{2} - \frac{\sin(\pi)}{2} = \frac{\pi}{2} - 0 = \frac{\pi}{2} \)
At lower bound \( x = 0 \): \( 0 - \frac{\sin(0)}{2} = 0 - 0 = 0 \)
Total: \( \frac{1}{2} \left( \frac{\pi}{2} - 0 \right) = \frac{\pi}{4} \)