Optimization problem involving discrete and continuous variables

Question

In my mechanics class I am to solve an optimization problem that is basically reduced to finding a set of points $x_i,y(x_i)$ such that the following functional is minimized: $$\sum\limits_{i}^{} \int_{f_1(x_{i-1},x_i)}^{f_2(x_i,x_{i+1})} f(x,x_i,y(x_i)) \ dx$$ Does anybody know what mathematical model can be used in this situation? I am familiar with the idea of optimizing a functional using Euler-Lagrange equations but here it is more difficult because the problem has discrete elements. After a bit of searching I came across the concept of Mixed Integer Nonlinear Programming. Would that be applicable here? Any help would be appreciated