I Partial Fraction Decomposition With Quadratic Term

1. Oct 1, 2016

Cosmophile

Hey, all! I'm learning partial fraction decomposition from Serge Lang's "A First Course in Calculus." In it, he gives the following example:

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

He then decomposes this into the following sum:

$$\frac{x+1}{(x-1)^2(x-2)} = \frac{c_1}{x-1}+\frac{c_2}{(x-1)^2}+\frac{c_3}{x-2}$$

My question is this: On the right hand side (RHS), $x-1$ and $(x-1)^2$ appear. Why is this the case, when the original denominator only had the $(x-1)^2$? I hope this makes sense, and any help here is greatly appreciated!

2. Oct 1, 2016

mathman

If you left out the x-1 denominator term, then the numerator for $(x-1)^2$ would be a+bx. The expression you are given is equivalent and is easier to integrate.

3. Oct 1, 2016

Cosmophile

I'm afraid I don't really understand. Could you explain more explicitly, or direct me to a good resource on this?

4. Oct 2, 2016

pwsnafu

I've answer this in a previous thread, so read that first and then you can ask more questions here.

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