The usual change of variable technique takes the form

$$t=g(x)$$

$$\mathrm{d}t = g'(x)\,\mathrm{d}x$$

However, sometimes it is useful to relate the new variable and the old one *implicitely*:

$$h(t)=g(x)$$

$$h'(t)\,\mathrm{d}t = g'(x)\,\mathrm{d}x$$

$t$ may be explicitely expressed in terms of $x$, and this would lead to the same results, if we remember the derivative of the inverse function

$$t=h^{-1}(g(x))$$

$$t\mathrm{d}t = (h^{-1})'(g(x))g'(x)\,\mathrm{d}x=\dfrac{1}{h'(h^{-1}(g(x)))}g'(x)\,\mathrm{d}x=\dfrac{1}{h'(t)}g'(x)\,\mathrm{d}x$$

$$h'(t)\,\mathrm{d}t = g'(x)\,\mathrm{d}x$$

but this may be much more awkward. It is just a question of ease