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?

  • 7
    $\begingroup$ It isn't. A primitive of $x'x''$ would be $\frac12(x')^2$. $\endgroup$ – user239203 Jun 6 '20 at 1:55
  • $\begingroup$ can you write what you are claiming explicitly? why not use integral and dx notation to keep it clear. $\endgroup$ – 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 : $$I= \dfrac {\dot x^2}2+C$$


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