00:01
So we have the integral of 2x squared minus 10x plus 4 over x plus 1 times x minus 3 quantity squared dx.
00:16
So this problem is going to illustrate how we use partial fraction decomposition to make an integral like this easier.
00:24
First, you want to notice that the numerator and denominator are polynomials.
00:28
Second you want to notice that the denominator has a lower is a higher order than the polynomial in the numerator.
00:36
So that means we're also the denominator's factor.
00:40
So we're pretty much ready to start the decomposition right now.
00:43
So we take 2x squared minus 10x plus 4 over x plus 1 times x minus 3 quantity squared.
00:52
And so you want to set up the form.
00:54
So we're going to take each factor and put a constant over it.
00:58
So a over x plus 1 plus b over x minus 3.
01:04
I don't need this, plus c over x minus 3 squared.
01:12
And so for repeated roots, you need to represent each power of them.
01:16
That's why i have that x minus 3 and the x minus 3 quantity squared.
01:22
All right.
01:23
So for b and c, we can use the heavyside cover -up method.
01:29
So for b, we need the root.
01:33
So the root was x equals three and the root for, no, not looking at b.
01:38
So we're looking at a and c.
01:40
So the reason is if we multiply both sides by x minus three squared, there'd be a factor of x minus three times b and a factor of x minus three squared times a.
01:49
And it would cancel on the left hand side.
01:51
So if we just plugged in that value, we would know what c was.
01:54
Okay, so we have the root, the root set x equals three and x equal, negative 1.
02:00
So for a, we're covering up x plus 1, so we're going to get 2 times negative 1 squared minus 10 times negative 1 plus 4 over, so x plus 1 is covered.
02:14
So we have x, so negative 1 minus 3 quantity squared.
02:20
So that's going to be 2 plus 10 plus 4 over 4 squared, which is 16.
02:29
Well, negative 4 squared, no, doesn't matter.
02:30
So we get 16 over 16 over 16.
02:32
That equals one...