For part i) I got the answer 1/((jw)^2 + 5jw +6)

For part ii)

I first consider input to be a unit impulse

Thus, Y(w)=H(w)F(w) and F(w)=1

yI(t)=-1/2pi integrate from -infinity to infinity (e^jwt)/(w^2 - 5jw - 6) dw

where yI(t) is the output when f(t) is a unit impulse

using complex contour integration,

I got yI(t) = 0 for t<0

and yI(t) = 2je^(-2t) - 3je^(-3t) for t>0

Then using y(t)=yI(t)*f(t)

I got y(t) = integrate from -infinity to infinity ( 2je^(-2(t-k)) - 3je^(-3(t-k)) )f(k)dk

For a similar example in my notes, it just stops at this step.

However, for this question I am not sure whether should I stop here as I noticed that I havent use the boundary conditions yet.

Please gives me some idea on this.