calculus - Integrating equations relating infinitely small changes

I've been studying classical mechanics recently and I have two (related) questions on calculus:

1. In the first chapter of R. Shankar's book Fundamentals of Physics, he derives $vdv = adx$ (in a time interval $[t, t + dt]$ as $dt \rightarrow 0$) from the definitions of $a$ and $v$. Then he writes:

   
   
   
   $$\int_{v_1}^{v_2} vdv = a\int_{x_1}^{x_2} dx$$
   
   
   

   for the situation when $v$ and $x$ change in a time interval $[t_1, t_2]$, and I don't quite understand how that follows (as the quantities are equal in the same time intervals, while the integration is with respect to $v$ and $x$; and $v$ as a function of $v$ is completely different than $v$ as a function of $t$).




2. How to justify the variable change done here? It probably flows from the answer to the first question, but maybe there's something more to be said.