**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