Question

Evaluate \( \int_0^{\frac{\pi}{2}} \sin^2 x \, dx \)

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

Hint:

For integrals involving trigonometric functions, use standard trigonometric identities to simplify the expression before integrating.

Answer 1

The Correct Option is D

Step 1: Use the identity for \( \sin^2 x \).
We can simplify \( \sin^2 x \) using the half-angle identity: \[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
Step 2: Substitute into the integral.
Substitute this identity into the integral: \[ \int_0^{\frac{\pi}{2}} \sin^2 x \, dx = \int_0^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} \, dx \]
Step 3: Integrate.
We now integrate term by term: \[ \frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx - \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(2x) \, dx \] The first term gives: \[ \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4} \] The second term is: \[ \frac{1}{2} \cdot \left[ \frac{\sin(2x)}{2} \right]_0^{\frac{\pi}{2}} = 0 \]
Step 4: Conclusion.
Thus, the value of the integral is \( \frac{\pi}{4} \).