# Why is it possible to do this

How is it possible that one can break up the derivative operator such as this:

$$\frac{dv}{dt}=t^2$$, then integrate like this,
$$\int^v_{v_{0}}dv = \int^t_0 t^2 dt$$, where$$v=v_{o}$$ when $$t=0$$. Especially in light of what most calculus teachers tell you; that the derivative symbol is not a fraction and should not be interpreted as a faction?

If f(t) = g(t) and f(t) is integrable, then $\int f = (\int g) + C$ where C is an undetermined constant. Your use of boundary conditions (v(0) = v0) allows you to determine the constant.
What you're really integrating on the right side is $$\int^v_{v_{0}} \frac {dv}{dt} dt$$