= Solution
For the critically damped case, causality reduces the <convolution> to
$$
y(t)=\int_0^\infty s e^{-\kappa s}\cos[\kappa(t-s)]\,ds.
$$
Although the forcing is not integrable on the full real line, this integral converges absolutely because the <Green's function> is integrable. Thus the calculation is valid directly; the corresponding <Fourier transform> can be interpreted as a <tempered distribution>.
Use the <complex exponential> and $\int_0^\infty s e^{-as}\,ds=a^{-2}$ for $\operatorname{Re}a>0$:
$$
y(t)=\operatorname{Re}\left[e^{i\kappa t}\int_0^\infty s e^{-\kappa(1+i)s}\,ds\right]
=\operatorname{Re}\left[\frac{e^{i\kappa t}}{\kappa^2(1+i)^2}\right].
$$
Since $(1+i)^2=2i$, the <convolution> response is
$$
\boxed{y(t)=\frac{\sin(\kappa t)}{2\kappa^2}}.
$$
This is the bounded steady response to the stated forcing acting for all time. Without an <initial condition> or the <convolution> selection, the general solution additionally contains $(C_1+C_2t)e^{-\kappa t}$. No switch-on at $t=0$ was specified, so zero initial displacement and <velocity> should not be imposed on this all-time forcing.
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