The transition from the 3rd equation to the 4th (the 1st being the one next to W=...) confuses me, what happens to the nabla, why does the derivative of the trajectory function become just the function? Is the integral part integrated between these equations? Elucidation greatly appreciated.
This is the chain rule. Reading from bottom up, differentiate $V$ with respect to time by differentiating it with respect to its argument and multiplying by the derivative of the argument with respect to time.
You're missing a $\mathrm dt$ at the end of the last integral.