The [x(1-x^r)^n] bit equates to zero when subbing in the boundaries, so it basically just disappears

Add/subtract trick comes down mostly to intuition. In this case, seeing that you have an nr out the front of the integral and x

^{r} just hanging around, with part of I

_{n-1} should tell you that you should manipulate this in some shape or form to the original, especially since you have an nr+1 coefficient for I

_{n}. Basically, you're looking for an nr x (I

_{n-1} - I

_{n}) to manipulate the integral to find the result. A good way of thinking about it is that if you have a result, think about what you're working towards and think about how you might get to that result. It really just comes down to practice and intuition