Integration of rational of polynomials

Question

I want to evaluate the indefinite integral for: $$ \int\frac{x^3+3x−2}{x^2-3x+2}dx,\quad \text{for } x>2 $$ I did long division and factoring, simplifying it to $$ \int x+3\,dx + \int\frac{10x-8}{(x-1)(x-2)}dx $$ The integral for the first part is easy: $\frac{x^2}{2}+3x+c$. It's the second part I'm having trouble with. I looked up the solution for that part and it says that the second part simplifies to: $$ \frac{12}{x-2}-\frac{2}{x-1} $$ which can easily be integrated as $12\ln(x-2)-2\ln(x-1)+c$, which can be added to the last partial answer for the complete answer. I can see that when you multiply the two fractions in the difference above by $x-2$ and $x-1$ respectively and combine them, you arrive at the more complicated expression above. My question is, what reasoning can I use to get from the fraction above to these two simpler fractions, and how can I identify this case in other problems?

Answer 1

You need the construction called partial fractions (Wiki link). The idea is that you want to write $$ \frac{10x-8}{(x-1)(x-2)} = \frac{A}{x-1} + \frac{B}{x-2} $$ so you can bring the RHS to common denominator and compare the numerators, getting 2 equations in 2 unknowns.

Answer 2

Write: $$\frac{10x-8}{(x-1)(x-2)}= {a\over x-1}+{b\over x-2}$$

then you have to find $a$ and $b$. That you do after clearing the denominators:

$$10x-8 = a(x-2)+b(x-1)$$ so you have to solve a system $10 =a+b$ and $-8=-2a-b$