Calculus/Work

$$\begin{align} W =\int F\,dr &=\int m a\,dr \\ &=\int m \, \frac{dv}{dt} \, dr \\ &=m \int \frac{dr}{dt} \, dv \\ &=m\int v \, dv \\ &=\frac{mv^2}{2} =\Delta E_k \end{align}$$