Derivation of Kinetic energy formula and energy principle

Hi all,

Here is the derivation of kinetic energy from Work:

W = ∫Fds
From the second law of motion F = dp/dt, which is equal to mdv/dt, so:

W = m∫dvdx/dt which = m∫dv x v because dx/dt = v
Therefore W = 1/2mv², when integrated.
However from simple algebra derivation, W = Δ1/2mv².

Did I skip something in the derivation? I know that the actual integration would be 1/2mv²
+ C, where C is a constant. However I omitted the C. Is it because of that that I did not end up with the proper equation? If so, why would C = -1/2v_i². (v_i because Δv = v - v_i and the rest comes from kinetic energy equation.) Or is it because I wrongly integrated?

Any answers would be appreciated. Thanks