Using the table and propeties of Laplace transform to find d

Using the table and propeties of Laplace transform to find (d) d^ {3/(s+2)^5 + s-2/s^2-25+5}

Solution

L^-1(3/(s+2)^5) = 3.L-1(1/(s+2)^5)

use frequency shifting property L-1(F(S-a)) = e^at

= .3.e^(-2t) L-1(1/S^5)

from correspondence table L-1(1/S^(n+1)) = t^n/n!

= 3.e^(-2t).L-1(1/S^5))

= 3.e^(-2t).(t^4/4!)

= 3.e^(-2t). t^4/24)

= t^4/8 . e^(-2t)

taking second term

L-1(S-2/(S^2-2S+5)) = L-1((S-2)/((S-2)^2+1))

= e^(-2t)cos(2t)

finally we get answer  = t^4 / 8 . e^(-2t)+e^(-2t)cos(2t)