What is the name of the theorem that allows one to change the variable of integration by multiplying the integrand by the variable’s derivative?

Question

I have often stated in my coursework without a second thought that

$$\int_{X} y \, dx = \int_T y \, \frac{dx}{dt}\,dt$$

What is the name of this theorem? Is there any specific way in which one should invoke it during a formal proof?

I believe it is probably the change of variables theorem,

$$\int_{u(a)}^{u(b)} f(u(x))\,d(u(x)) = \int_a^b f(u(x)) \, \frac{d(u(x))}{dx} \, dx$$

but I genuinely do not know.

Answer 1

Yes, is change of variable. The article of wolfram that you cite, is exactly what you're asking for.