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}
<br /> V_c(s)[s^2+2s+2]+2=\frac{s}{s^2+1}<br />
so i get
<br /> V_c(s)=\frac{s}{(s^2+2s+2)(s^2+1)}-\frac{2}{s^2+2s+2}<br />
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?