Convolution to compute the following Inverse Laplace Transform

Question

Use the method of Convolution to compute the following Inverse Laplace Transform and simplify the final answer as much as possible:

Answer 1

Decompozition give us: s²/(s² + 1)² = - 1/(s² + 1)² + 1/(s² + 1).

So, L^-1{s²/(s² + 1)²} = L{- 1/(s² + 1)²} + L^-1{1/(s² + 1).} = - 1/2sint + 1/2tcot + sint =

1/2(sint + tcost)