Chapter 1, Problem 1.32P

Show that

${\displaystyle \int (2ar\cdot \stackrel{\dot{}}{r}+2b\stackrel{\dot{}}{r}\cdot \ddot{r})dt=a{r}^2+b{\stackrel{\dot{}}{r}}^2+\text{const}\text{.}}$

where r is the vector from the origin to the point (x₁, x₂, x₃). The quantities r and $\stackrel{\dot{}}{r}$ are the magnitudes of the vectors r and $\stackrel{\dot{}}{r}$, respectively, and a and b are constants.