A rarely seen form of substitution of definitite integral theorem: $\int_\alpha^\beta f(u(t))dt=\int_{u(\alpha)}^{u(\beta)}f(x)\cdot(u^{-1})'(x)dx$

Question

Below is a theorem about substitution of definite integral that I found today. However, I had never seen this form in analysis books. How to understand its meaning and usage? Is it really made used in practice? The characters for $u^{-1}$, $f(u(t))$ is messy to me. (Though I know the classic form of such theorem.)

Let $J=[\alpha,\beta],~u:J\to\Bbb R$ be a $C^1$ function and $u'(x)\neq 0$ for all $x\in J$, $I$ be an interval and $u(J)\subseteq I$, $f:I\to\Bbb R$ be continuous. Then $$\int_\alpha^\beta f(u(t))dt=\int_{u(\alpha)}^{u(\beta)}f(x)\cdot(u^{-1})'(x)dx$$

Edit:

We are looking for explicitly examples where this method is usefully for computing integrals.