When integrating by $u$-substitution, instead of declaring a $u$ variable explicitly, the substitution can also be done inline like this:
$$ \int \! \sin x \cos x \, \textrm{d}x = \int \! \sin x \cos x \, \frac{\textrm{d}(\sin x)}{\cos x} = \int \! \sin x \, \textrm{d}(\sin x) = \frac{\sin^2 x}{2} + C $$
Is there a name for this approach to $u$-substitution? Also, might there be a reason to use or not use this approach? For example, might it be more error prone for some reason?