MIT exercise 6.003 HW2 - Concept of system initially at rest
The mistake is in the final integration of the term

$$\frac{1}{2i}\big(e^{it}-e^{-it}\big)u(t)=\sin(t)u(t)\tag{1}$$

because you seem to ignore the step function $u(t)$. Integrating $(1)$ gives

$$\int_{-\infty}^t\sin(\tau)u(\tau)d\tau=u(t)\int_0^t\sin(\tau)d\tau=\big[1-\cos(t)\big]u(t)\tag{2}$$