Calcule \int \cos^2(x) \, dx. a) \frac{x}{2} + \frac{\sin(2x)}{4} + C b) \sin^2(x) + C c) \tan(x) + C d) \frac{x}{2} + C

Para calcular a integral \(\int \cos^2(x) \, dx\), podemos usar a identidade trigonométrica: \[ \cos^2(x) = \frac{1 + \cos(2x)}{2} \] Assim, a integral se torna: \[ \int \cos^2(x) \, dx = \int \frac{1 + \cos(2x)}{2} \, dx = \frac{1}{2} \int (1 + \cos(2x)) \, dx \] Agora, calculamos a integral: \[ \frac{1}{2} \left( \int 1 \, dx + \int \cos(2x) \, dx \right) = \frac{1}{2} \left( x + \frac{\sin(2x)}{2} \right) + C \] Portanto, temos: \[ \int \cos^2(x) \, dx = \frac{x}{2} + \frac{\sin(2x)}{4} + C \] A alternativa correta é: a) \(\frac{x}{2} + \frac{\sin(2x)}{4} + C\)