Consider the Laplace transform Xs1ss2 decide if x exists or

Consider the Laplace transform X(s)=1/s(s+2), decide if x() exists or not, and find it if it does.

Solution

X(s)=1/s(s+2)

L^-1[X(s)]=L^-1(1/s(s+2))=A/s +B/(s+2) partial differentiation method

=A(s+2)+B(s)=1

s=-2 B=-1/2

s=0 A=1/2

A/(s) +B/(s+2)=1/2s -1/(s+2)2 =1/2 u(t)-1/2 e^-2t =x(t)

x(infinity)=1/2u(infinity)=(1/2)*1-0=1/2

e^(-infinity)=0