Proving existence and uniqueness of solutions of this IVP: $x'(t)=f(x(t))+\sin(t)$, with $x(0)=0$ and $f\in C^2$

Question

I'm doing this exercise:

Let $f$ be a $C^2$ function such that $\|f\|_\infty\leq M$. Prove that \begin{cases}x'(t)=f(x(t))+\sin(t) \\ x(0)=0\end{cases} has an unique solution defined for all $t\in \mathbb{R}$.

What we can tell about $f(x(t))+\sin(t)$ is that it's locally Lipschitz, but that only gives us locally uniqueness.

I've already proved that solutions have to be defined for all $t \in \mathbb{R}$, but I don't know how can we show uniqueness.

Thanks for your time.