Use the definition and properties of Laplace transforms to find the Laplace transform of each of the following functions:

(a)

\[y_{1}(t)=t e^{-a t}+e^{-a t} \cos \omega t\]

(b)

\[y_{2}(t)=t e^{-2 t} \cos 3 t+t^{2} \sin 2 t\]

(c)

\[y_{3}(t)=\frac{d}{d t}\left(t^{2} e^{-t}ight)+\int_{0}^{t} \lambda^{2} e^{-\lambda} d \lambda\]