How to pick $u$ and $v^{\prime }$?

1. 1)

   Whatever $v^{\prime }$ is, you need to be able to integrate it to find $v$.

2. 2)

   $u^{\prime }$ should be simpler (or no more complicated) than $u$

3. 3)

   $v$ should be simpler (or no more complicated) than $v^{\prime }$

For example, we very often want $u(x)$ to be a polynomial if that’s an option, because $u^{\prime }(x)$ will have lower degree than $u(x)$ and hence be simpler.