$(20$ points$)$ Laplace Transform

$($a$)$ Find the Laplace transforms of the following signals and determine their region of convergence.

i$.(5$ points$)f(t)=t{e}^{-at}(sin{}_{0}t{)}^{2}u(t)$

ii$.(5$ points$)f(t)=e^{-b|t|}$ where $b0$

$($b$)$ The Laplace transform of a causal signal $x(t)$ is given by

$x(s)=\frac{1}{s^2+2s+5},$ROC:$Re\{s\}>-1$

Which of the following Fourier transforms can be obtained from $x(s)$ without actually determining the signal $x(t)?$ In each case, either determine the indicated Fourier transform or explain why it cannot be determined.

i$.(5$ points$)F\{x(t)e^{\frac{t}{2}}\}$

ii$.(5$ points$)F\{x(t)e^{2t}\}$