When we use a substitution $u = g(x)$ to compute an indefinite integral $\int f(x) dx$ or a definite integral $\int_a^b f(x)dx$, what is required of the function $g$ in each case? Here are what I think. Please correct me if I am wrong.
For some integrals, we need to express $x$ in terms of $u$; in that case, I think we should require $g$ to be invertible on $\mathbb{R}$ (in the case of $\int f(x) dx$), or on $[a, b]$ (in the case of $\int_a^b f(x)dx$). Or else we need to discuss by cases, i.e. break the function $g$ into invertible pieces.
For integrals that do not require us to express $x$ in terms of $u$, consider for example $\int \sin^2 x \cos x dx$ and $\int_0^{\frac{2\pi}{3}} \sin^2 x \cos x dx$. Here $u = \sin x$ would work, but notice that the function $g(x) = \sin x$ is not invertible on $[0, \frac{2\pi}{3}]$, let alone on $\mathbb{R}$. So it seems to me that invertibility is not necessary.
In short, my question is about precisely which functions $g$ yield legitimate substitutions $u = g(x)$.
$g(x)$ only has to be such that it is differenetiable.
See UCSD notes.
I suppose there is also assumption that $g'(x)$ is an integrable function.