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The usual change of variable technique takes the form

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

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

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

$t$ 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

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

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