Assume that the system is underdamped, starts from rest, and the force is bounded (|f(t)| < A for all t). Given the formula for the oscillator: $$ x(t)= \frac{1}{\beta m}\int_{0}^{t}f(t-v)e^{-b/2m}sin\beta v dv. $$ Where $$ \beta = \frac{1}{2m}\sqrt{4mk-b^{2}} $$ Show that $$ \left | x(t) \right | \leq \frac{2A}{\beta b} $$
Obviously, we need to manipulate the expressions inside the integral to reduce to x(t) to the desired fraction. I know that we could simplify f(t-v) inside the integral to A since |f(t)| < A for all t. Beyond that, I'm not sure what the next steps are.