The area bounded by the curve $y=f\left(x\right)$ the coordinate axes and the line x = t is given by $t{e}^t$ then $f\left(x\right)=$

$\mathrm{Area} \mathrm{bounded} \mathrm{by} \mathrm{the} \mathrm{curve} \mathrm{y}=\mathrm{f}\left(\mathrm{x}\right) ,$

$\mathrm{the} \mathrm{coordinate} \mathrm{axes} \mathrm{and} \mathrm{the} \mathrm{line} \mathrm{x} = \mathrm{t}$

$\mathrm{is}{\int }_0^{\mathrm{t}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} ={\mathrm{te}}^t$

$$\begin{gathered} \Rightarrow f\left(t\right)=e^t\left(t+1\right) \\ \therefore f\left(x\right)=e^x\left(x+1\right) \end{gathered}$$