Take into account that is so that...
I have attached an example from my notes for contour integration.
However I don't understand how they derive the parametrized equation
I would appreciate if someone could please explain this to me
If you're wondering how they came up with I would say that they wrote the example that way. There's nothing to derive with respect to the choice of contour. There's loads of freedom in deciding on contours over which to integrate, depending on the situation.