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

L^-1{s²/(s² + 1)²} = L^-1{s/(s² + 1)} • L^-1{s/(s² + 1)} = cost • cost = ∫₀^tcosvcos(t - v)dv = 1/2∫₀^t(cost + cos(2v - t))dv = 1/2(vcost + 1/2sin(2v - t))₀^t = 1/2tcost + 1/4sint - 1/4sin(-t) = 1/2(tcost + sint);

here • is convolution sign.