Hi,
Can anyone explain how this follows? Has it been factories and simplified?
Thanks!
Yes thanks, it's all quite straight forward from there. I think I need to revisit algebraic long division though it's been about 3 years since I've used it!
RE the method used directly above, in general when given:
A/B = (C.B)/B - D/B = C - D/B ???
This bears some resemblance to completely the square?
Here is a trick: let $\displaystyle y = 1+t$ so that $\displaystyle y-1 = t$, then:
$\displaystyle \begin{aligned} & \frac{2t^3}{1+t} = \frac{2(y-1)^3}{y} = \frac{2 y^3-6 y^2+6 y-2}{y} = \\& 2y^2-6y+6-\frac{2}{y} = 2(1+t)^2-6(1+t)\\&+6-\frac{2}{1+t} = 2t^2-2 t+2-\frac{2}{t+1}.\end{aligned} $
Or, better yet, use the substitution as an integral substitution.