can someone help with the steps underlined, how do you i get from one to the other?
Thanks.
When you have $\displaystyle \frac{u^2}{u^2-k}$
this can be rearranged to a form which can be directly integrated
using the tables of standard known integrals
by adding and subtracting k in the numerator, whatever constant k is.
$\displaystyle \frac{u^2}{u^2-k}=\frac{u^2-k+k}{u^2-k}=1+\frac{k}{u^2-k}$
both of which which can easily be integrated.
More generally, just do a "long division"
$\displaystyle u^2- 0u- 4)\overline{u^2+ 0u+ 0}$
Obviously the "$\displaystyle u^2$" will divide into the "$\displaystyle u^2$" 1 time so we will then need to subtract $\displaystyle (u^2+ 0u+ 0)- (u^2- 0u- 4)= 4$. $\displaystyle u^2- 4$ divides into $\displaystyle u^2$ 1 time with remainder 4: $\displaystyle \frac{u^2}{u^2- 4}= 1+ \frac{4}{u^2- 4}$.