I don't see any danger in reading x_i as x(i) analogous to f(x)... so x is your vertical/output axis (usually y or f(x)), i your horizontal, input axis (instead of x), then proceed (and integrate) as normal.
This problem comes up quite frequently in economics (and I am completely stuck...)
I have the choice to spread my consumption between a continuum of goods (indexed by over the interval [0,N]. Let be how much of good that I consume. Let be my total consumption (the total consumption of each individual good). We can express this as .
Notice that we are integrating with respect to , not . I have no idea how to manipulate the above expression. Can we just take the anti-derivative with respect to ? How could I solve for an individual ? Any advice (no matter how little), would be greatly appreciated.