00:03
Okay, for this integral, the first part we have to do is to figure out how that denominator factors.
00:11
There's multiple ways of doing that.
00:13
You can plug it into your graph and calculator and look for where your x intercepts are to find the factors.
00:19
You can hopefully notice that in each pair there's a common factor, and so you can factor this by grouping.
00:27
You can pull out some tricks from algebra 2 or college algebra about the rational zero theorem, all different ways of figuring out how this factors.
00:39
And this fraction actually factors into 1 over x plus 2 times x minus 2 squared.
00:49
And so when we break this down, we're going to have a linear factor of x plus 2, so a over x plus 2.
00:58
And we have the repeated linear factor of x minus 2.
01:02
So we need two fractions.
01:03
We need a b over x minus 2 to the first power, and then a c over x minus 2 to the second.
01:15
So there's the form of our decomposition.
01:19
We need to multiply everything by our common denominator, and that's the x plus 2 and the x minus 2 squared.
01:28
And so each fraction, when we multiply it, something's going to be dividing out, and we have to take our numerator times the stuff that's left over.
01:38
So on the left -hand side, everything divides out, and we just get one.
01:43
So i'm going to have a times the x minus two squared.
01:48
So you'd have to foil that and take a times that answer, which would be a times x squared, and then minus a times 4x.
02:01
So that'll give us a 4ax, and then a times the 4.
02:06
So plus 4a.
02:09
When we have the second fraction with the b in it, one of the x minus 2s will divide out.
02:15
So i've got x plus 2 times x minus 2 times my b, which is just x squared minus 4 times b.
02:24
So that'll give me plus bx squared minus 4b.
02:30
And then for the c, both of the x minus 2s divide out, but i still have the x plus 2.
02:36
So c times that gives me plus cx plus 2c...