From frequent commenter, SRK:

*A question for commenters: how to explain / teach integration by substitution? To organise discussion, consider the simple case *

* Here are some options.*

*1) Let . This gives , hence . So our integral becomes . Benefits: the abuse of notation here helps students get their integral in the correct form. Worry: I am uncomfortable with this because students generally just look at this and think “ok, so dy/dx is a fraction cancel top and bottom hey ho away we go”. I’m also unclear on whether, or the extent to which, I should penalise students for using this method in their work.*

*2) Let . This gives . So our integral becomes . Benefit. This last equality can be justified using chain rule. Worry: students find it more difficult to get their integral in the correct form.*

*3) has the form where and . Hence, the antiderivative is . This is just the antidifferentiation version of chain rule. Benefit. I find this method crystal clear, and – at least conceptually – so do the students. Worry. Students often aren’t able to recognise the correct structure of the functions to make this work.*

*So I’m curious how other commenters approach this, what they’ve found has been effective / successful, and what other pros / cons there are with various methods.*

**UPDATE (21/04)**

Following on from David’s comment below, and at the risk of splitting the discussion in two, we’ve posted a companion WitCH.