Today one of my colleagues asked me how this followed: Thinking back my response (in front of the class) was something along the lines of “that’s a fairly standard trick used when integrating”. This troubles me as it makes it seem that it has come out of nowhere by some kind of magical process and probably adds to the feeling that mathematics is a just a collection of rules to be followed.
A better description, in my mind would be something like the following: “When faced with an integral that you can’t do straight away, try to manipulate it to something that you can do by either ‘adding 0’ or ‘multiplying by 1’. That is you do something that changes the form of the integrand but which leaves it equivalent to what you started with. In this case you ‘add 0’ by adding 4 and then subtracting 4:”
In my experience professional mathematicians use this phrase to mean a “commonly used technique” and so it doesn’t have the connotations that the phrase can have.
Should we avoid the phrase? Or try to push for a “maths specific interpretation”?