[tex]\delta(x)[/tex] has the property that for any continuous [tex]f[/tex] with support in [tex]\mathbb R[/tex],
[tex]\displaystyle\int_{-\infty}^\infty f(x)\delta(x-c)\,\mathrm dx=f(c)[/tex]
where [tex]c\in\mathbb R[/tex]. So we have
[tex]\displaystyle\int_{-\infty}^\infty e^{5t}\delta(t-2)\,\mathrm dt=e^{10}[/tex]
[tex]\displaystyle\int_{-\infty}^\infty\cos2t\,\delta(t-5)\,\mathrm dt=\cos10[/tex]
The remaining two integrals are identical to the Laplace transforms of [tex]\cos3t\,\delta(t-2)[/tex] and [tex]t^3\sin t\,\delta(t-3)[/tex]. For these, we have the property that
[tex]\mathcal L_s\{f(t)\delta(t-c)\}=f(c)e^{-sc}[/tex]
so we get
[tex]\displaystyle\int_0^\infty e^{-st}\cos3t\,\delta(t-2)\,\mathrm dt=\mathcal L_s\{\cos3t\,\delta(t-2)\}=\cos6\,e^{-2s}[/tex]
[tex]\displaystyle\int_0^\infty e^{-st}t^3\sin t\,\delta(t-3)\,\mathrm dt=\mathcal L_s\{t^3\sin t\,\delta(t-3)\}=27\sin3\,e^{-3s}[/tex]
