Question 9.14: Find the inverse transform of the following function F(s) = ......

This is an improper function since the orders of the numerator and the denominator are equal. We begin by performing a long division in order to change the function into a quotient and a remainder

which yields

F(s) = 1 + \frac{−3s^{2}  −  11s  −  5}{s^{3}  +  6s^{2}  +  11s  +  6} = 1 + \frac{−3s^{2}  −  11s  −  5}{(s  +  1)(s  +  2)(s  +  3)}

We can expand the right-hand equation using partial fractions resulting in

F(s) = 1 + \frac{\frac{3}{2}}{s  +  1} + \frac{-5}{s  +  2} + \frac{\frac{1}{2}}{s  +  3}

The inverse transform is

f(t) = δ(t) + \left(\frac{3}{2} e^{-t}  –  5e^{-2t} + \frac{1}{2} e^{-3t}\right) u(t)