Foorier transformation question

[electron2]

this is the last part of a bigger question
if you i forgot to mention some data please tell

in the last part i got
$\ddot{v_c}+2\dot{v_c}+2v_c=v_s(t)$
$\dot{v_c(0)}=2$
$v_c(0)=0$

i need to find out if the response for $v_s(t)=u(t)+cos(t)$
equals the sum of $v_c^u$(the response for spet function) and $v_c^cos$ the response for cosine
i start by finding the response for cos(t)
$v_c(t)->V_c(s)$
$\dot{v_c}->sV_c(s)-v_c(0)$
$\ddot{v_c}->s(sV_c(s)-v_c(0))-\dot{v_c(0)}$
$s(sV_c(s)-v_c(0))-\dot{v_c(0)}+2sV_c(s)-v_c(0)+2V_c(s)=\frac{s}{s^2+1}$
$V_c(s)[s^2+2s+2]+2=\frac{s}{s^2+1}$
so i get
$V_c(s)=\frac{s}{(s^2+2s+2)(s^2+1)}-\frac{2}{s^2+2s+2}$
where $\dot{v_c}=j\omega v_c$

so now i need to break the fractures into a simpler ones
but here its all complex and i don't know how to get
a simpler fractures and there foorier transformation
??

so this is where i got stuck
and i even didnt get closer to the main solution of the problem

??

 

[maverick280857]

First of all, unless you put in $s = j\omega$, the s-domain transformation is known as a Laplace Transform. The Fourier Transform is a special case of the Laplace Transform with $s = j\omega$.

As for your query, do you know how to express the right hand side in terms of a partial fraction expansion, in terms of functions of s whose inverse Laplace transform (time domain function) is known to you?