I'm working on a physics assignment and am having some trouble. I need to integrate $r\dot\theta^2$ with respect to $t$. However, my trouble lies in the definition of the upper-dot format.
Given: $$ \dot\theta=\frac{\mathrm{d}}{\mathrm{d}t}\theta $$ and $$ \ddot\theta=\frac{\mathrm{d}^2}{\mathrm{d}t^2}\theta $$
If I square $\dot\theta$, I get: $$\dot\theta^2 = \left(\frac{\mathrm{d}}{\mathrm{d}t}\theta\right)\left(\frac{\mathrm{d}}{\mathrm{d}t}\theta\right)=\frac{\mathrm{d}^2}{\mathrm{d}t^2}\theta^2$$
Does this mean that: $$\dot\theta^2 = \ddot\theta\theta$$
So if I go back to my original task: $$\int r\dot\theta^2\, \mathrm{d}t \rightarrow \int r\ddot\theta\theta \,\mathrm{d}t$$ Which I still don't really know how to integrate, my first thoughts are a combination of chain rule and integration by parts, any help is appreciated!