Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it.
\begin{equation}\frac{1}{((s+\frac{1}{2})^2+\frac{1}{4})*s}\end{equation}
This 1/s is confusing me, my solution is:
\begin{equation}-2cos(\frac{t}{2})+2e^{\frac{-t}{2}}+2sin(\frac{t}{2})\end{equation}
Convolution in time domain corresponds to multiplication in the s-domain. You can decompose the laplace transform as $$ \frac{1}{((s+1/2)^2+1/4)s} = \frac{1}{((s+1/2)^2+1/4)}\cdot\frac{1}{s} $$ The second term corresponds to the laplace transform of the step function $u(t)$. The first term is $$ \frac{1}{((s+1/2)^2+1/4)} $$ The inverse laplace transform of this function should not be difficult. Let's call it $f(t)$.
Then you will have the convolution of $f(t)$ and $u(t)$ in the time domain. This would correspond to a function $h(t)$ such that $h(t)=0$ for $t\leq 0$, and then for $t>0$ $$ h(t) = \int_0^tf(x)d(x). $$