Why integration of $\dot{x}\ddot{x}$ is equal to $4\dot{x}^2$? [closed]

I know how to calculate the basic integral such as $$x$$ or $$x^2$$ but I don't understand why the integral of $$\dot{x}\ddot{x}$$ is equal to $$4\dot{x}^2$$.

I know this is a very elementary question but can someone teach me the integration process?

• It isn't. A primitive of $x'x''$ would be $\frac12(x')^2$. – user239203 Jun 6 '20 at 1:55
• can you write what you are claiming explicitly? why not use integral and dx notation to keep it clear. – jimjim Jun 6 '20 at 2:09

Integrate $$\dot x \ddot x$$ w.r.t the variable $$t$$: $$I=\int \dot x \ddot x dt=\int \dot x \dfrac {d\dot x}{dt}dt=\int \dot x d \dot x$$ Finally you get : $$I= \dfrac {\dot x^2}2+C$$