Question

Evaluate the integral: \[ \int_0^\pi \sin^2(x) \, dx \]

• A. $\frac{\pi}{2}$
• B. $\frac{\pi}{4}$
• C. $\pi$
• D. 1

Hint:

Use trigonometric identities to simplify integrals involving squared trigonometric functions.

Answer 1

The Correct Option is A

Use the identity $\sin^2(x) = \frac{1 - \cos(2x)}{2}$: \[ \int_0^\pi \sin^2(x) \, dx = \int_0^\pi \frac{1 - \cos(2x)}{2} \, dx \] \[ = \frac{1}{2} \int_0^\pi 1 \, dx - \frac{1}{2} \int_0^\pi \cos(2x) \, dx \] \[ = \frac{1}{2} \left[ x \right]_0^\pi - \frac{1}{4} \left[ \sin(2x) \right]_0^\pi \] \[ = \frac{\pi}{2} - 0 = \frac{\pi}{2} \]