\[ \begin{align*} \mathscr{L}\left\{ t\right\} & =\intop_{0}^{\infty}te^{-st}\,\mathrm{d}t\\ & \xrightarrow{\text{IBP}}\left[\begin{aligned}u & =t & \mathrm{d}v & =e^{-st}\,\mathrm{d}t\\ \mathrm{d}u & =1\,\mathrm{d}t & v & =\frac{e^{-st}}{-s} \end{aligned} \right]\\ & =u*v-\int v\,\mathrm{d}u\\ & =t\frac{e^{-st}}{-s}+\frac{1}{s}\intop_{0}^{\infty}e^{-st}\,\mathrm{d}t\\ & =t\frac{e^{-st}}{-s}+\frac{1}{s}\left[\frac{e^{-st}}{-s}\right]_{0}^{\infty}\\ & =t\frac{e^{-st}}{-s}+\frac{1}{s^{2}} \end{align*} \]