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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?

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Hint: Take the derivative of both sides with respect to $t$.

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It's an application of substitution. We have that: $$ \int u \; \frac{du}{dt} \; dt = \int u \; du $$ Now replace $u$ by $\frac{dx}{dt}$.

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With $$u=\frac {dx}{dt}$$ we get $$\frac {du}{dt}=\frac {d^2x}{dt ^2}$$

The integral becomes $$\int udu=\frac {1}{2}u^2$$

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