Qual é a integral definida \( \int_0^{\pi/2} \cos^2(x) \, dx \)? A) \( \frac{\pi}{4} \) B) \( \frac{\pi}{2} \) C) \( \frac{1}{2} \) D) \( 0 \)

Para calcular a integral definida \( \int_0^{\pi/2} \cos^2(x) \, dx \), podemos usar a identidade trigonométrica: \[ \cos^2(x) = \frac{1 + \cos(2x)}{2} \] Substituindo na integral, temos: \[ \int_0^{\pi/2} \cos^2(x) \, dx = \int_0^{\pi/2} \frac{1 + \cos(2x)}{2} \, dx \] Isso se separa em duas integrais: \[ = \frac{1}{2} \int_0^{\pi/2} 1 \, dx + \frac{1}{2} \int_0^{\pi/2} \cos(2x) \, dx \] Calculando a primeira integral: \[ \frac{1}{2} \int_0^{\pi/2} 1 \, dx = \frac{1}{2} \left[ x \right]_0^{\pi/2} = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4} \] Agora, a segunda integral: \[ \frac{1}{2} \int_0^{\pi/2} \cos(2x) \, dx = \frac{1}{2} \left[ \frac{\sin(2x)}{2} \right]_0^{\pi/2} = \frac{1}{2} \cdot \frac{1}{2} \left( \sin(\pi) - \sin(0) \right) = 0 \] Portanto, somando as duas partes: \[ \int_0^{\pi/2} \cos^2(x) \, dx = \frac{\pi}{4} + 0 = \frac{\pi}{4} \] Assim, a resposta correta é: A) \( \frac{\pi}{4} \)