To integrate the Lorentz factor over a path where the velocity is changing, you need to consider the relativistic effects of time dilation and length contraction. The Lorentz factor is given by:
γ = 1 / sqrt(1 - (v/c)^2)
where γ is the Lorentz factor, v is the velocity of an object, and c is the speed of light.
To integrate the Lorentz factor over a path, you would need to determine the velocity as a function of position along the path. If the velocity is changing between the two points, you will need to express the velocity as a function of position, such as v(x) or v(t) depending on whether the path is described by position or time.
Once you have the velocity function, you can integrate the Lorentz factor over the path using the appropriate integration variable. For example, if the path is described by position (x), you would integrate γ dx, and if the path is described by time (t), you would integrate γ dt.
The specific method of integration will depend on the form of the velocity function. If the velocity is a simple function, you can use standard integration techniques. However, if the velocity function is more complex, you may need to use numerical methods to perform the integration.
It's worth noting that integrating the Lorentz factor over a changing velocity path can be a challenging task analytically, and it may require numerical techniques or approximation methods in many cases.