You can derive this without any calculus methods. From the Galileo's equation we have: $vf^2=vi^2+2ad$, where vf is the final velocity, vi the initial velocity, a the acceleration produced by a constant force and d the distance. Multiply the equation with $m/2$ and we got $mvf^2/2-mvi^2/2=2mad$. But $2mad$ is the work $W$. So $mvf^2/2-mvi^2/2=W$. Also from the fundamental work-energy theorem we have $Ef-Ei=W$, where Ef,Ei are the energies coresponding to the final,initial states. So in accordance to that, we deduce that $KE=mv^2/2$

You can derive this without any calculus methods. From the Galileo's equation we have: $v_f^2=v_i^2+2ad$, where $v_f$ is the final velocity, $v_i$ the initial velocity, a the acceleration produced by a constant force and d the distance. Multiply the equation with $m/2$ and we got $mv_f^2/2-mv_i^2/2=2mad$. But $2mad$ is the work $W$. So $mv_f^2/2-mv_i^2/2=W$. Also from the fundamental work-energy theorem we have $Ef-Ei=W$, where Ef,Ei are the energies corresponding to the final,initial states. So in accordance to that, we deduce that $KE=mv^2/2$