The line integral along this path is the integral of f(p(t)).p′(t). In other words, the integrand is the dot product of the force field and the velocity vector. We are integrating with respect to time, as time runs from start to end, i.e. the time interval that traces the path. Obviously p must be differentiable, at least piecewise differentiable, else the line integral is not defined. The derivative p′ is the velocity, how fast the object is traveling at each point in time. The dot product asks whether the object is traveling with, against, or perpendicular to the field. The dot product increases with the speed of the object and the strength of the field. The line integral measures the energy involved in tracing the path. Moving against the field consumes energy, while moving with the field returns energy to the object. Moving perpendicular to the field gives a dot product of 0 and contributes nothing to the line integral, like walking along the side of a mountain, moving neither up hill nor down hill.
The line integral through a linear combination of fields is equal to the linear combination of the individual line integrals, traveling through each field separately. Also, the line integral across an entire path is the sum of the line integrals along the subpaths, provided the subpaths join together to make the entire path. All this follows from the properties of integrals.
Furthermore, it doesn't matter how fast we traverse the path. We can even stop for a while, back up, go forward again, go fast, go slow, etc. This is called a reparametrization. Let u(s) be a continuous, piecewise differentiable map from an interval in s onto (start to finish) the interval in t that describes our path. If s is the new measure of time, The line integral is f(p(u(s))) dotted with the derivative of p(u(s)), which brings in u′(s) by the chain rule. But aha, this is merely integration by substitution; the result is the same.