Find the inverse Laplace Transform of the function

Question

Find the inverse Laplace Transform of:

G(s) = (as+b) / (s^2+β^2)

Answer 1

We know

L{sin(βt)} = β/(s²+β²) and L{cos(βt)} = s/(s²+β²)

Thus

g(t) = L^-1{(as+b)/(s²+β²)} = aL^-1{s/(s²+β²)} + bL^-1{1/(s²+β²)} = acos(βt)+bsin(βt)/β

Answer 2

G(s) = (as + b)/(s² + β²) = as/(s² + β²) + b/(s² + β²);

L^-1(G(s)) = L^-1(as/(s² + β²)) + L^-1(b/(s² + β²)) = acos(βt) + b/βsin(βt)