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When doing a multiple integral it is often helpful/necessary to use a change of variables.

In some cases the change is guided by making the region of the integration simpler - for example, turning a parallelogram into the unit square.

In others, it is guided by making the integrand itself simpler, which is always the goal in single-integration.

I was wondering if there are any general principles for deciding between these two priorities. Also, is changing variables in multiple integration just generally "harder"? U-substitutions are obviously useful to me, but I can't shake off the feeling that most of the substitution examples in my multivariable calc book are somewhat contrived... how much is it actually used in practice?

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My opinion:

"Most" integrals don't have closed forms. While changing variables is a useful technique, the only reason it always works in textbooks is because they have been contrived to allow you to do so - or you are working with a very basic function (eg volume of a sphere).

So you have to start with the assumption that the integral is solvable by a change of variables. This limits down the possible integrals that you are looking at solely to ones that 'magically' simplify - and with that knowledge, it's normally pretty obvious what the 'magical' change of variables was contrived to be.

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