00:01
Let's evaluate the given integral.
00:03
Before we do partial fraction to composition, we should see if we could factor the denominator.
00:09
So let's rewrite that numerator.
00:13
And then on the bottom, we can take out of x.
00:17
We get x squared plus 3.
00:20
We still have a quadratic down here.
00:23
So we'd like to know whether or not that can be factored because if it can be factored, it must be factored before you do partial fractions.
00:31
So here we look at the discriminant b squared minus 4 ac where these are coming from the coefficients of your polynomial.
00:45
In our case we see a is 1.
00:47
There's no x term so b has to be 0 and then we see c is 3.
00:54
So plugging this in we have 0 squared minus 4 times 1 times 3.
01:00
That's definitely less than 0.
01:02
So this will tell us that this quadratic does not factor.
01:08
So let me come up here to write the next part.
01:11
When we do the partial fraction to composition, for the x, it's not repeated.
01:18
It's a linear factor.
01:20
So that gives us a over x.
01:24
And then for the next term, we have a quadratic doesn't factor.
01:29
So we have bx plus c.
01:32
I'm writing out of room up here.
01:34
This is me to go back a few steps.
01:41
So we have a over x and then bx plus c.
01:44
That's a linear factor because we have a quadratic on the bottom.
01:50
So to see where the bx plus c comes from in the text, this is what the author calls case 3...
