Partial fraction decomposition

[HorseBox]

Homework Statement

$$\frac{2x + 2}{x^2 - 2x + 1}dx$$

Homework Equations

The Attempt at a Solution

I factorized the denominator and got $$\frac{2x + 2}{(x - 1)^2}$$ and I took a look at the solutions manual to see how they handled this but then I see this $$\frac{2x + 2}{(x - 1)^2} = \frac{A}{(x - 1)} + \frac{B}{(x - 1)^2}$$. I'm lost how is the denominator of partial fraction B squared?

 

[elect_eng]

Homework Statement

$$\frac{2x + 2}{x^2 - 2x + 1}dx$$

Homework Equations

The Attempt at a Solution

I factorized the denominator and got $$\frac{2x + 2}{(x - 1)^2}$$ and I took a look at the solutions manual to see how they handled this but then I see this $$\frac{2x + 2}{(x - 1)^2} = \frac{A}{(x - 1)} + \frac{B}{(x - 1)^2}$$. I'm lost how is the denominator of partial fraction B squared?

That is the rule used when you have degenerate poles (factored terms in the denominator with powers greater than one). The simplest way to understand this is to try and find an equality without following this rule. You will not be able to do it.


Take a look at the following link and note the section on repeated factors in the denominator. The use of this rule ensures that you have enough degrees of freedom to solve for the coefficients in all cases.

http://en.wikipedia.org/wiki/Partial_fraction