Consider the constraint of the form:
$\sum \limits_{i}A_i\dot{x}_i + B = 0$ where $i=1,2,3$
In general, this equation is nonintegrable.
But if $A_i = \frac{\partial f}{\partial x_i}$,$B = \frac{\partial f}{\partial t}$ and $f = f(x_i,t)$
Then the equation can be rewritten as: $\sum \limits_{i}\frac{\partial f}{\partial x_i}\frac{dx_i}{dt} + \frac{\partial f}{\partial t} = 0$
which is: $\frac{df}{dt} = 0$
and integrated to: $f(x_i,t) -$ constant $ = 0$
Why is it in general nonintegrable?
From what I understand, as $A_i$ and $B$ can be functions of any variables that are in the constraint, thus we may not be able to integrate the constraint so that we get a constraint consisting of non-$\dot{x}_i$ variables. These differential constraints may not be exact.
Look here for example: https://physics.stackexchange.com/questions/482949/non-integrable-differential-equation-and-non-holonomic-contraints/719438#719438