Let $f\left(x\right)=\underset{0}{\overset{x}{\int }}{e}^{t}f\left(t\right)dt+e^x$ be a differentiable function for all $x\in R.$ Then $f\left(x\right)$ equals:

$\begin{array}{rcl}\mathrm{f}\left(\mathrm{x}\right)& =& {\int }_0^{\mathrm{x}} {\mathrm{e}}^{\mathrm{t}} \mathrm{f}\left(\mathrm{t}\right) \mathrm{dt}+{\mathrm{e}}^{\mathrm{x}}\\ {\mathrm{f}}^1\left(\mathrm{x}\right)& =& {\mathrm{e}}^{\mathrm{x}} \mathrm{f}\left(\mathrm{x}\right)+{\mathrm{e}}^{\mathrm{x}}\\ & =& {\mathrm{e}}^{\mathrm{x}}\left(\mathrm{f}\left(\mathrm{x}\right)+1\right)\\ \int \frac{{\mathrm{f}}^1\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)+1}& =& \int {\mathrm{e}}^{\mathrm{x} }\mathrm{dx}\\ \log \left(\mathrm{f}\right(\mathrm{x})+1)& =& {\mathrm{e}}^{\mathrm{x}}+\mathrm{c}\\ \mathrm{since} \mathrm{f}\left(0\right)& =& {\int }_0^0 {\mathrm{e}}^{\mathrm{t}} \mathrm{f}\left(\mathrm{t}\right) \mathrm{dt}+{\mathrm{e}}^0\\ & =& 1\\ & \Rightarrow & \log \left|\mathrm{f}\left(0\right)+1\right|={\mathrm{e}}^0+\mathrm{c}\\ \log 2-1& =& \mathrm{c}\\ & & \therefore \log \left|\mathrm{f}\right(\mathrm{x})+1|={\mathrm{e}}^{\mathrm{x}}+\log 2-1\\ \log \frac{\left|\mathrm{f}\right(\mathrm{x})+1|}{2}& =& {\mathrm{e}}^{\mathrm{x}}-1\\ \frac{\left|\mathrm{f}\right(\mathrm{x})+1|}{2}& =& {\mathrm{e}}^{({\mathrm{e}}^{\mathrm{x}}-1)}\end{array}$