Let $X : [t_0, t_f] \rightarrow R_{\geq 0}$ be a set of nonnegative piecewise continuous functions with a finite number of discontinuities defined on an interval $[t_0, t_f]$ and such that satisfy a constraint of the type $\frac{dx}{dt} = f(x(t))$, where $f$ is a continuous function of $x$. Let $x^* \in X$ be a solution to the following optimization problem $$ \max_{x \in X} \int_{t_0}^{t_f} e^{x(t)}dt$$ I want to understand if I can make the assumption that $x^*$ is simultaneously the solution to the following optimization problem: $$ \max_{x \in X} \int_{t_0}^{t_f} x(t)dt$$
I made this assumption in my work, because I wanted to avoid having an integral of the exponential in my set of ODEs (to avoid stiffness). However, reviewing my work, I realized that I never really showed that this substitution can be done, am not actually sure it is valid. I am not sure how to go about proving it. I apologize if the problem is not properly stated, or if I didn't use the appropriate tags - my mathematical background is somewhat lacking.