Differentials and Integration
Ng Tze Beng
The problem with the change of variable in integration is the trade off between convenience and understanding. It is essentially a consequence of the chain rule. One should put the integral in the following form
and seek the antiderivative F of f . Then the integral is given by F(g(x)) + C. That is
It is convenient to write y = g(x). Then we have . Then it is preferably to proceed in the following manner:
, where F is an antiderivative of f
The use of the differential in this formula is all too pervasive and lack any genuine mathematical sense than the suggestive convenience of cancellation of dx. Although it can be justified in most cases but the price to pay is costly because of the need to establish a meaning of integration over the differential 'dx' .
The following article by Allendoerfer sums up the situation very well.
A/Prof. Ng Tze Beng