Integrate $ \int dr(dr+2r)\left(1-\frac{r+\frac{dr}{2}}{r_0}\right) $

Question

How can I integrate the following?

$$ \int dr(dr+2r)\left(1-\frac{r+\frac{dr}{2}}{r_0}\right) $$

Where $r_{0}$ is a constant and $r=[0, r_{0}]$

Answer 1

Notice, $$\int dr(dr+2r)\left(1-\frac{r+\frac{dr}{2}}{r_0}\right)$$ $$=\int ((dr)^2+2r\ dr)\left(1-\frac{2r+dr}{2r_0}\right)$$ as $dr\to 0$ hence, neglecting $(dr)^2$ $$=\int 2r\ dr\left(\frac{2r_0-2r-dr}{2r_0}\right)$$

$$=\frac{1}{r_0}\int r\left(2(r_0-r)dr-(dr)^2\right)$$ neglecting $(dr)^2$

$$=\frac{1}{r_0}\int 2r(r_0-r)dr$$ $$=\frac{1}{r_0}\int 2rr_0\ dr-\frac{1}{r_0}\int 2r^2\ dr$$ $$=2\int r\ dr-\frac{2}{r_0}\int r^2\ dr$$