Why is Kinetic Energy mv^2/2?

Whilst this proof is beyond the scope of A level physics, it is well within the scope of A level Maths as it relies solely on the chain rule.First let us note that Fx = W where F is force, x is distance and W is work or energy.However if we have a varying force, we must sum the Force in dx sized chunks, ie. W = int ( F dx) and it is from there we get our proof.F = ma (from Newton II)a = dv/dtF = mdv/dtPutting this in our integral:int(Fdx) = int (m * dvdx/dt)We also know that v = dx/dt.Therefore int(mdvdx/dt) = int(mdvv) = int(mv dv).Taking the integral we can see that W = mv²/2 + C where C is dependant on the initial speed we take the work done from. If initial speed is 0 then C also becomes 0 and we get the well known formula.