Problema 25: Determine o valor de \( \int_0^{\pi/2} \cos^2(x) \, dx \). a) \frac{\pi}{4} b) \frac{1}{2} c) \frac{\pi}{8} d) \frac{3\pi}{8}

Para resolver a integral \( \int_0^{\pi/2} \cos^2(x) \, dx \), 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 pode ser separado 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: \[ \frac{1}{2} \int_0^{\pi/2} 1 \, dx = \frac{1}{2} \cdot \left[ x \right]_0^{\pi/2} = \frac{1}{2} \cdot \frac{\pi}{2} = \frac{\pi}{4} \] Agora, a segunda integral: \[ \frac{1}{2} \int_0^{\pi/2} \cos(2x) \, dx = \frac{1}{2} \cdot \left[ \frac{\sin(2x)}{2} \right]_0^{\pi/2} = \frac{1}{2} \cdot \left( \frac{\sin(\pi)}{2} - \frac{\sin(0)}{2} \right) = \frac{1}{2} \cdot 0 = 0 \] Portanto, somando os resultados: \[ \int_0^{\pi/2} \cos^2(x) \, dx = \frac{\pi}{4} + 0 = \frac{\pi}{4} \] Assim, a alternativa correta é: a) \(\frac{\pi}{4}\)