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$$ \frac{d}{dx} (\frac{1}{2} v^2) = v \frac{dv}{dx} $$

$$ \int \ \frac{d}{dx} (\frac{1}{2} v^2) \ dx = \frac{1}{2} v^2 $$

$$ \int \ v \frac{dv}{dx} \ dx = \frac{1}{2} v^2 $$