Comment by gus_massa

[I teach Math in the first year of the university in Argentina. We have a few Calculus courses, with different levels according to the degree.]

In 1D, substitution by linear functions like "t=3x+1" is very insightful. It's a pity that sometimes we don't have time to analyze it more deeply. Other substitutions may be insightful or not. Some tricks like "t=sin(x)" has a nice geometrical interpretation, but it's never explained, we don't teach it anyway now.

Integration by parts is not very insightful until you get to the 3rd or 4th year and learn Solovev spaces or advanced Electrodynamics. I'd like to drop it, but other courses require it and I'd be fired.

In some cases, parity and other symmetries are interesting, but those tricks are mostly teach in Physics than in Math.

Also, in the second year we get 2D or 3D integrals, that have a lot of interesting variable changes. Also, things like the Gauss theorem and it's relation with conservation laws.