We complete the square on the denominator first:


`Lap^{:-1:}{3/((s+2)^2+3^2)}=e^(-2t) sin 3t`

`g(t)=e^(-2t) sin 3t`

(The boundary curves `f(t)=e^(-2t)` and `f(t)=-e^(-2t)` are also shown for reference.)

0.511.522.5-0.50.51-0.5-1tg(t)Open image in a new page

Graph of `g(t)=e^(-2t) sin 3t`.

For interest: Here's the Scientific Notebook answer:


`=-sqrt(-36)/12("exp"((-2+sqrt(-36)/2)t)` `{:-"exp"((-2-sqrt(-36)/2)t))`


This answer involves complex numbers and so we need to find the real part of this expression.

`"Re"(-j/2(e^((-2+3j)t)-e^((-2-3j)t)))` `=e^(-2t)sin 3t`