Question

Find the value of the integral \( \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx \).

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{6} \)

Hint:

Use the identity for \( \sin^2(x) \) to simplify the integral and break it into simpler parts for easier evaluation.

Answer 1

The Correct Option is A

The objective is to evaluate the integral: \[ I = \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx. \]

Applying the identity \[ \sin^2(x) = \frac{1 - \cos(2x)}{2}, \] 

the integral transforms to: \[ I = \int_0^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} \, dx = \frac{1}{2} \int_0^{\frac{\pi}{2}} (1 - \cos(2x)) \, dx. \] 

Decomposing the integral yields: \[ I = \frac{1}{2} \left[ \int_0^{\frac{\pi}{2}} 1 \, dx - \int_0^{\frac{\pi}{2}} \cos(2x) \, dx \right]. \] 

The individual integrals are evaluated as: \[ \int_0^{\frac{\pi}{2}} 1 \, dx = \frac{\pi}{2}, \quad \int_0^{\frac{\pi}{2}} \cos(2x) \, dx = 0. \] 

Consequently, the integral's value is: \[ I = \frac{1}{2} \left( \frac{\pi}{2} - 0 \right) = \frac{\pi}{4}. \]