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

The inverse Laplace transform of $$1/\left( {{s^2} + 2s} \right)$$ is

The Laplace transform of the function
$$\eqalign{ & f\left( t \right) = k,\,0 < t < c \cr & \,\,\,\,\,\,\,\,\, = 0,\,c < t < \infty ,\,\, \cr} $$
is

The Laplace Transform of a unit step function $${u_a}\left( t \right),$$ defined as
$$\matrix{ {{u_a}\left( t \right) = 0} & {for\,\,\,t < a\,} \cr { = 1} & {for\,\,\,t > a,} \cr } $$ is