Minimum Integral Exponential Constraint

Let $\mathcal{F}$ be the set of continuous functions $f : \mathbb{R} \to \mathbb{R}$ which satisfy the condition $$ e^{f(x)} + f(x) \geq x + 1,$$ for any real number $x$. Find the minimum value attained by the integral $$ I(f) = \int_{0}^{e} f(x) \, dx, $$ for $f$ lying in $\mathcal{F}$.