I have heard the comparison between line integrals and work, so how would I minimize the absolute value of the line integral of two points over a non-conservative vector field (if it were conservative, the line integral would be constant). I understand something like the Euler Lagrange Equation is needed to find a function that minimizes the absolute value of the line integral, however I am confused as to how to set up the Lagrangian and how to evaluate the equation.

Take the following example:

Find a function that minimizes the absolute value of the line integral between (-1, 2) to (2, 8) over the vector field $F=\langle y^2 \cos(x), \sin(xy)\rangle$.