44. Qual é o valor de \( \int_{0}^{\pi/2} \cos^2(x) \, dx \)? a) \( \frac{\pi}{4} \) b) \( \frac{\pi}{2} \) c) \( \frac{\pi}{3} \) d) \( \frac{\pi...

Para resolver essa integral, podemos usar a identidade trigonométrica \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). Substituindo na integral, temos: \[ \int_{0}^{\pi/2} \cos^2(x) \, dx = \int_{0}^{\pi/2} \frac{1 + \cos(2x)}{2} \, dx \] \[ = \frac{1}{2} \int_{0}^{\pi/2} 1 \, dx + \frac{1}{2} \int_{0}^{\pi/2} \cos(2x) \, dx \] \[ = \frac{1}{2} \left[x\right]_{0}^{\pi/2} + \frac{1}{2} \left[\frac{\sin(2x)}{2}\right]_{0}^{\pi/2} \] \[ = \frac{1}{2} \left[\frac{\pi}{2} - 0\right] + \frac{1}{2} \left[\frac{\sin(\pi)}{2} - \frac{\sin(0)}{2}\right] \] \[ = \frac{\pi}{4} + \frac{1}{2} \times 0 \] \[ = \frac{\pi}{4} \] Portanto, a alternativa correta é: a) \( \frac{\pi}{4} \)