Step 1 : Understanding the Question:
This problem asks us to compute the value of a definite integral involving the square of the sine function over the interval from \( 0 \) to \( \pi/2 \).
Step 2 : Key Formulas and Approach:
Integrating power-based trigonometric terms is simplified by using double-angle identities to reduce the degree of the integrand. The power-reduction identity for sine is:
\[ \sin^2(x) = \frac{1 - \cos(2x)}{2} \]
Alternatively, we can use the definite integral symmetry property (King's Property):
\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx \]
Step 3 : Detailed Explanation:
Step 1: Substitute the power-reduction identity into the definite integral:
\[ \int_{0}^{\pi/2} \sin^2(x) \, dx = \int_{0}^{\pi/2} \frac{1 - \cos(2x)}{2} \, dx \]
Step 2: Factor out the constant term \( \frac{1}{2} \) to simplify the integrand:
\[ = \frac{1}{2} \int_{0}^{\pi/2} (1 - \cos(2x)) \, dx \]
Step 3: Find the antiderivative of the terms inside the integral:
\[ = \frac{1}{2} \left[ x - \frac{\sin(2x)}{2} \right]_{0}^{\pi/2} \]
Step 4: Evaluate the antiderivative at the upper limit \( x = \pi/2 \):
\[ \frac{1}{2} \left( \frac{\pi}{2} - \frac{\sin(\pi)}{2} \right) = \frac{1}{2} \left( \frac{\pi}{2} - 0 \right) = \frac{\pi}{4} \]
Step 5: Evaluate the antiderivative at the lower limit \( x = 0 \):
\[ \frac{1}{2} \left( 0 - \frac{\sin(0)}{2} \right) = 0 \]
Subtracting the lower limit evaluation from the upper limit evaluation yields:
\[ \frac{\pi}{4} - 0 = \frac{\pi}{4} \]
Step 4 : Final Answer:
The value of the definite integral is \( \frac{\pi}{4} \), which corresponds to option (A).