Question about substitution method in integration

It is common that we replace $\int u(x)v′(x)\mathrm{d}x$ by $\int u \mathrm{d} v$ where both $u$ and $v$ are continuous functions of $x$. My question is, must we ensure that $u$ can be written as a function of $v$ before applying this? The above substitution method is involved in the proof of integration by parts but I cannot find textbooks that addressed this point.

I find it easier to understand for definite integrals because I can think it as summation. But for indefinite integrals it is just anti-derivative and I am not sure how this kind of replacement is valid.

In terms of differential, of course $\mathrm{d}v$ and $v′(x) \mathrm{d}x$ are the same. But
$∫$
and
$\mathrm{d}x$
together forms one single mathematical sign meaning anti-derivative for indefinite integral,

$∫f(x)\mathrm{d}x$
means finding the primitive function $F(x)$ which is a function of $x$. So to me
$∫u\mathrm{d}v$
implies $u$ is function of $v$.

I just wonder why in applications substitution is done in this way without considering the differentiability of $u$ with respect to $v$.