After some head scratching yes it does, finaly.
I think my mistake has always been trying to integrate wrt time. I really had to step back and ask myself what i was after, work that is, which is not a time integral.
Manipulating dv/dt to dx/dt and then subbing in V now makes it...
I have a question that annoys my basic understanding of kinetic energy.
I know if I have a force-time plot then the area under the curve is equivalent to the impulse imparted on an object (in units Newton-Seconds). I know that this is also equivalent to the change in momentum of the object...