Given a function $u\colon \Bbb R_+ \to \Bbb R$ with the following boundedness property $$\forall t\in \Bbb R_+,\ \exists u_0>0;\ |u(t)|\le u_0,$$ consider the following integral; $$I(t)=\int_{0}^{t}\sin \left( w(t-\tau )\right) u(\tau )~{\rm d}\tau.$$ Now, the question is, when $I(t)$ is bounded and if it is, how to find an estimate $I_0$ of the bound so that $|I(t)|\le I_0$? Moreover, $I_0\propto u_0$?
Note. My guess is that $I(t)$ is bounded iff the frequency spectrum of $u(.)$ does not involve $w$. But have no idea how to find $I_0$.