Why can we make this integral change of limits? Is it obvious? When deriving the equation for the impulse-momentum theorem, the following occurs:
$$\cdots=\int\limits_{t_1}^{t_2}\frac{d\vec p}{dt}dt = \int\limits_{\vec p_1}^{\vec p_2}d\vec p=\cdots$$
I know the $dt$s aren't simply canceling each other out, so why can we get rid of them and then why do the limits of integration change? Is it because once the $dt$s are gone the limits just change automatically since we're only left with $d\vec p$ or is there something that needs to be done and it just happens that the limits change because of it?
Thank you for your time. 

This is the chain rule.  Sometimes the chain rule is stated as $\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}$, just as if $du$ literally refers to a number and cancels.
$$
\int_a^b \frac{dp}{dt}\,dt = \int_{t=a}^{t=b} f'(p(t))p'(t)\,dt \tag 1
$$
where $f'$ is a constant function equal to $1$.  So $f(p)=p+\text{constant}$.  So by the chain rule $(1)$ is
$$
\int_{t=a}^{t=b} (f\circ p)'(t)\,dt = f(p(b))-f(p(a)) = \int_{p=p(a)}^{p=p(b)} f'(p)\,dp = \int_{p=p(a)}^{p=p(b)} 1\,dp.
$$
