Let f(t) be piecewise continuous function on [0, infinity) and is of exponential order a (i.e. | f(t) | 0). a) Show that laplace transform:

Let f(t) be piecewise continuous function on [0, infinity) and is of exponential order a (i.e. | f(t) | <= M e^{\alpha t} , M > 0).

a) Show that laplace transform: L\{ \{f(t)}{t} \ } = int_{0}^{infinity} F(u) du , when s > \alpha

b) Use the result in part (a) to find the laplace transform: L\{ {sinh(t)}/{t} }

c) Use the result in part (a) to find inverse laplace: L{ {(s+1)}{(2s^{2} + 4s + 5)^{2}}