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The process of evaluating a line integral involves summing up the product of a vector function with the differential element along a curve. This kind of integration takes into account not just the function's value at certain points, but how those values change as you move along the path defined by the curve.

Let's consider the example provided. To evaluate the line integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\), we break down the path C into simple segments that we can work with. In this case, two line segments. The integral along each piece can be thought of as a separate problem, yet they are connected by their position along the path. We must ensure that each segment is evaluated with respect to the parametrization of the curve.

The parametrization of a curve is a way to describe a curve by specifying the coordinates as functions of one or more parameters. It's like providing a set of instructions or a map that tells you how to trace out the curve in space as you change the parameter values. A good parametrization will make evaluating the line integral simpler and more intuitive.

In our case, we broke the path into two segments and defined r₁(t) and r₂(t) as the parametric equations representing each segment of the curve. Parametrizations convert complicated paths into simpler, straight-line motions that are easier to handle. Specifically, we used the parameter t to trace each segment of the path, which then guided the evaluation of the line integrals over the defined intervals.

The Fundamental Theorem of Line Integrals states that, under certain conditions, the line integral of a gradient field along a path depends only on the values of the function at the endpoints of the path. This remarkable theorem simplifies calculations and offers deep insights into the nature of conservative vector fields.

Although the theorem isn't directly applied in the step-by-step solution to our problem, it's vital to understand the conditions under which the theorem can be applied, as it has the power to turn a seemingly hard integral into a straightforward substitution. When dealing with vector fields that are conservative, the Fundamental Theorem of Line Integrals enables us to avoid parametrization and directly evaluate the function at the end points. However, when a vector field isn't conservative, as in the given exercise, we must return to the more traditional method of line integral evaluation through parametrization and careful integration.