(i) Unit Ramp Function

(r\left(t\right)=t\ \ \ \ t>0 )

(r\left(t\right)=0\ \ \ t<0 )

$\begin{align*} L\left\{\ r\left(t\right)\right\}=\int_0^{\infty{}}\begin{array}{l}r\left(t\right)e^{-st}\ dt\ \\ \ \end{array}=\int_{0^-}^{\infty{}}\begin{array}{l}te^{-st}\ dt\ \\ \ \end{array} =\frac{1}{s^2} \end{align*} $

(ii) Unit Impulse function

$\delta{}\left(t\right)=0\ \ \ t\not=0$

$\displaystyle and\ \int_{-\infty{}}^{\infty{}}\ \delta{}\left(t\right)dt=1\ \ \ \ t=0 $

$ \displaystyle L\left\{\delta{}\left(t\right)\right\}=\int_{0^-}^{\infty{}}\delta{}\left(t\right)e^{-st}\ dtt=1\ $