University Calculus: Early Transcendentals (3rd Edition)

$W=\dfrac{1}{2}mv_{2}^2-\dfrac{1}{2}mv_{1}^2$
Newton's second law of kinematics is defined as: $F=\dfrac{mdv}{dt}$ $W=\int_{x_1}^{x_2} \dfrac{mdv}{dt} dx$ Thus, we have $\int_{x_1}^{x_2} \dfrac{mdv}{dt} dx=\int_{x_1}^{x_2} (mv) \dfrac{dv}{dx} dx$ or, $\int_{x_1}^{x_2} (mv) \dfrac{dv}{dx} dx=\int_{v_1}^{v_2} (mv) dv$ or, $W=[\dfrac{mv^2}{2}]_{v_1}^{v_2}=\dfrac{1}{2}mv_{2}^2-\dfrac{1}{2}mv_{1}^2$ Hence, it has been proved.