What is #int (x^3-2x^2+6x+9 ) / (-x^2- x +3 )#?
First, we must decide what we want to do. Whenever you see a rational function (i.e. a polynomial divided by a polynomial), you want to use partial fraction decomposition (PFD).
We're going to take out a negative sign from the denominator and bring it back after the integral. Therefore, we want to be able to integrate
The first step in PFD is always to have the upper degree (currently 3) be lower than the lower degree (2). Therefore, we can start long dividing our polynomials:
This is a lot easier to manage!
Now we can use PFD. First, we need to factor the bottom. Using the quadratic formula, we find the denominator has roots at
We will call these
Using a little bit of algebra, we solve
Therefore, we have taken the original and made it much more manageable (well at least when it uses a bunch of variables to make it clear)
Now we can finally solve!
Or writing it out,