Let
[tex]F(s) = L\left\{e^{\lambda t} \sin(\omega t)\right\} = \dfrac{\omega}{(s-\lambda)^2 + \omega^2}[/tex]
By the result of ex. 6,
[tex]-F'(s) = L\left\{t e^{\lambda t} \sin(\omega t)\right\} = \boxed{\dfrac{2\omega (s-\lambda)}{\left((s-\lambda)^2 + \omega^2\right)^2}}[/tex]
In a similar way, you'll find that
[tex]L\left\{t e^{\lambda t} \cos(\omega t)\right\} = \boxed{\dfrac{(s-\lambda)^2 - \omega^2}{\left((s-\lambda)^2 + \omega^2\right)^2}}[/tex]
