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Calculating Moment Of Inertia

The problem I am working on is: A uniform, thin, solid door has height 2.10 m, width 0.835 m, and mass 24.0 kg. (a) Find its moment of inertia for rotation on its hinges. (b) Is any ...
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Showing $ m\int \frac{d\textbf{v}}{dt} \dot \normalsize \textbf{v}dt = \frac{m}{2}\int \frac{d(v^2)}{dt}{}dt$

Can someone please explain how this equation is valid, using intermediate steps if available? $$ m\int \frac{d\textbf{v}}{dt} \dot \normalsize \textbf{v}dt = \frac{m}{2}\int \frac{d(v^2)}{dt}{}dt$$ ...