# Is $2\int\frac{d^2x}{dt^2}*\frac{dx}{dt}dt=(\frac{dx}{dt})^2$ true??

I was watching a video online about motion under inverse square law here and the producer mentioned that, $$2\int\frac{d^2x}{dt^2}*\frac{dx}{dt}dt=\left(\frac{dx}{dt}\right)^2$$ i donot understand why is that so. I searched online but i didn't find anything! Can someone explain it to me please?

Hint: Take the derivative of both sides with respect to $$t$$.
It's an application of substitution. We have that: $$\int u \; \frac{du}{dt} \; dt = \int u \; du$$ Now replace $$u$$ by $$\frac{dx}{dt}$$.
With $$u=\frac {dx}{dt}$$ we get $$\frac {du}{dt}=\frac {d^2x}{dt ^2}$$
The integral becomes $$\int udu=\frac {1}{2}u^2$$