Para calcular a integral \(\int \sin^2(x) \, dx\), podemos usar a identidade trigonométrica: \[ \sin^2(x) = \frac{1 - \cos(2x)}{2} \] Assim, a integral se torna: \[ \int \sin^2(x) \, dx = \int \frac{1 - \cos(2x)}{2} \, dx = \frac{1}{2} \int (1 - \cos(2x)) \, dx \] Agora, integramos: \[ = \frac{1}{2} \left( x - \frac{1}{2} \sin(2x) \right) + C \] Portanto, a integral é: \[ \frac{1}{2}x - \frac{1}{4}\sin(2x) + C \] Analisando as alternativas: a) \(\frac{1}{2}x - \frac{1}{4}\sin(2x) + C\) - Correta b) \(-\frac{1}{2}x + \frac{1}{4}\sin(2x) + C\) - Incorreta c) \(\frac{1}{2}x + \frac{1}{4}\sin(2x) + C\) - Incorreta d) \(\sin(x) + C\) - Incorreta A alternativa correta é: a) \(\frac{1}{2}x - \frac{1}{4}\sin(2x) + C\).