The Laplace transform of a function $f(t)$ defined on $[0,)$ is defined to be$,$

$L\{f(t)\}(s)={\displaystyle \underset{0}{\overset{}{}}}f(t)e^{-st}dt.$

The result is a function whose domain is the set of all $s$ values for which this improper integral

converges.

$($a$)$ Show that $L\{sin(t)\}=\frac{1}{s^2+1}.$

$($b$)$ Show that $L\{\frac{f(t)}{t}\}={\displaystyle \underset{s}{\overset{}{}}}L\{f(t)\}()d.$

$($c$)$ Use parts $($a$)$ and $($b$)$ to show that ${\displaystyle \underset{0}{\overset{}{}}}\frac{sin(t)}{t}dt=\frac{}{2}.$