Laplace transform of $$f\left( x \right) = \cos \,h\left( {ax} \right)$$ is

The Laplace transform of a function $$f(t)$$ is $$$F\left( s \right) = {{5{s^2} + 23s + 6} \over {s\left( {{s^2} + 2s + 2} \right)}}$$$
As $$t \to \propto ,\,\,f\left( t \right)$$ approaches

A delayed unit step function is defined as $$$u\left( {t - a} \right) = \left\{ {\matrix{ {0,} & {t < a} \cr {1,} & {t \ge a} \cr } } \right.$$$

Its Laplace transform is ____________.

If $$L$$ denotes the laplace transform of a function, $$L\left\{ {\sin \,\,at} \right\}$$ will be equal to