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

Question

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?

Answer 1

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 : $$I= \dfrac {\dot x^2}2+C$$