Is this a quadratic factor or a repeated linear factor?

Question

If I need to integrate something like $\frac{1}{y^2(1-y)}$, and I use the method of partial fractions, how do I know whether the $y^2$ in the denominator is a quadratic factor or just a repeated linear factor. In other words, should i split it up like 1) $\frac{Ay+B}{y^2} + \frac{C}{1-y}$ or 2) $\frac{A}{y} + \frac{B}{y^2} + \frac{C}{1-y}$

Answer 1

The two formulas are completely identical--try combining $\frac{A}{y}+\frac{B}{y^2}$ into a single fraction to see it--so it really doesn't matter which one you use, mathematically. Practically, just use whichever of the two will make it easier to find an antiderivative.

Answer 2

you will get $$\frac{1}{y^2(1-y)}={y}^{-2}- \left( -1+y \right) ^{-1}+{y}^{-1}$$