I've noticed in my physics textbook (and in a lot of other popular sources), that the process of integrating non-constant acceleration to get to a velocity formula, the integrating bounds imposed on the velocity-part seem wrong.
In the above snippet, they're using $v_0$ and $v$ as bounds in the definite integral on the left-hand side of 3-34. Why though? Why not use the (IMO) more appropriate bounds of $0$ and $t$? If I'm not mistaken, according to the fundamental theorem of calculus, that definite integral would evaluate to:
$$\int_{v_0}^{v}\mathrm{d}v_x(t) = \bigl[v_x(t)\bigr]_{v_0}^{v} = v_x(v)-v_x(v_0) \tag{1}$$
Which doesn't make much sense to me. However, using $t$ and $0$ as the bounds, we get:
$$\int_{0}^{t}\mathrm{d}v_x(t) = \bigl[v_x(t)\bigr]_{0}^{t} = v_x(t)-v_x(0) = v_x(t) - v_0 \, \, \text{(by definition of $v_x(0)=v_0$)}$$
As people way smarter than me all seem to write it as in (1), I'm probably the one who's wrong. But why?
And furthermore, why are we even integrating between $v$ and $v_0$? Shouldn't we be integrating between $t$ and $0$ as velocity is, in this scenario, derived from acceleration (and thus we're really just looking at the total area underneath an $a$-$t$ graph)?