Extremizing an Integral which is the RHS of a Differential Equation

Question

Consider the problem of finding a continuously twice-differentiable function $x(t)$ which extremizes the convergent improper integral \begin{equation} I=\int_{-\infty}^{t} f(x,s)\mathop{ds} \end{equation} given that \begin{equation} G\left(x(t),\frac{\mathop{dx}}{\mathop{dt}},\frac{\mathop{d^2x}}{\mathop{dt^2}}\right)=I. \end{equation} Letting $X(s)=x(s)+\epsilon\eta(s)$, for continuously twice-differentiable $\eta(s)$ such that $\eta(t)=\lim_{s\rightarrow -\infty}\eta(s)=0$, \begin{align} I(\epsilon)&=\int_{-\infty}^{t} f(X,s)\mathop{ds},\\ I'(\epsilon)&=\int_{-\infty}^{t} \frac{\partial f}{\partial X}\frac{\partial X}{\partial \epsilon}\mathop{ds},\\ I'(0)&=\int_{-\infty}^{t} \frac{\partial f}{\partial x}\eta(s)\mathop{ds},\\ &=0. \end{align} Then \begin{align} I(\epsilon)&=G\left(X(t),\dot{X},\ddot{X}\right),\\ I'(\epsilon)&=\frac{\partial G}{\partial X}\eta_t+\frac{\partial G}{\partial \dot{X}}\dot{\eta}_t+\frac{\partial G}{\partial \ddot{X}}\ddot{\eta}_t,\\ I'(0)&=\frac{\partial G}{\partial x}\eta_t+\frac{\partial G}{\partial \dot{x}}\dot{\eta}_t+\frac{\partial G}{\partial \ddot{x}}\ddot{\eta}_t,\\ &=\frac{\partial G}{\partial \dot{x}}\dot{\eta}_t+\frac{\partial G}{\partial \ddot{x}}\ddot{\eta}_t,\\ &=0. \end{align} How can I proceed from here to derive an appropriate modification of the Euler-Lagrange equations for this case?