After reviewing change of variables, I realized that every text I read provided the change of variable equations in the problem descriptions. From what I understand, the main use of changing variables before integrating is to make a difficult integral into an easier integral. Problems given in textbooks or online give you a set of transformation equations that happen to nicely transform difficult integrals into integrals of relatively simple functions over relatively simple regions. If I were given a difficult looking integral "in the wild", I would have little idea about how come up with transformation equations that effectively simplify the calculations.
Suppose I am given a double or triple integral of a particular two or three variable function over a known region. For simplicity, suppose that the function and region are real-valued. How can I come up with a set of transformation equations that transform the given region into a specific region?
As examples, how could I transform any 2D region into a circle or a rectangle. In 3D, how could I transform any 3D region into a sphere or rectangular prism. These examples assume that a two or three variable function and 2D or 3D region are given.
I am guessing that simplifying the transformed region does not necessarily simplify the transformed function for integration. So, additionally, I would appreciate advice on intuiting what region to transform to for a given function and region.