Problema 14: Calcule a integral ∫_0^{π/2} cos^2(x) dx. a) π/4 b) π/6 c) π/3 d) π/2

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