The Dirac-delta function $\left ( \delta \left ( t-t_{0} \right ) \right )$ for $\text{t}$, $t_{0} \in \mathbb{R}$, has the following property

$$\int_{a}^{b}\varphi \left ( t \right )\delta \left ( t-t_{0} \right )dt=\left\{\begin{matrix} \varphi \left ( t_{0} \right ) & a< t_{0}< b\\ 0 & \text{otherwise} \end{matrix}\right.$$

The Laplace transform of the Dirac-delta function $\delta \left ( t-a \right )$ for $a> 0$;

$\mathcal{L}\left ( \delta \left ( t-a \right ) \right )=F\left ( s \right )$ is

1. $0$
2. $\infty$
3. $e^{sa}$
4. $e^{-sa}$