laplace transform

i have just started knowing laplace transfrom

but i dont have much matheamtical background in it

just using to solve electrical circuits

but while coming across the laplace transform of some time domain fuctions

i am noticing some pattern in which they come

for eg
\mathcal{L} ( e^{-at} ) =\frac{1}{s+a} \\ \mathcal{L} (\sin \omega t ) =\frac{\omega}{s^2+\omega^2} \\ \mathcal{L} ( \sin \omega t e^{-at} ) =\frac{1}{(s+a)^2+\omega^2} \\

but ot able to genralise its paatern

can some sirs help me with the general form of the

\mathcal{L} ( f(t)g(t))