Para calcular o valor de \( \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx \), podemos usar a identidade trigonométrica: \[ \sin^2(x) = \frac{1 - \cos(2x)}{2} \] Assim, a integral se torna: \[ \int_0^{\frac{\pi}{2}} \sin^2(x) \, dx = \int_0^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} \, dx \] Separando a integral, temos: \[ = \frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx - \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(2x) \, dx \] Calculando a primeira parte: \[ \frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4} \] Agora, para a segunda parte: \[ \int_0^{\frac{\pi}{2}} \cos(2x) \, dx = \left[ \frac{\sin(2x)}{2} \right]_0^{\frac{\pi}{2}} = \frac{\sin(\pi)}{2} - \frac{\sin(0)}{2} = 0 \] Portanto, a integral total é: \[ \frac{\pi}{4} - 0 = \frac{\pi}{4} \] Assim, a alternativa correta é: a) \( \frac{\pi}{4} \)