00:01
So let's look at the integral of the function x squared plus 2x plus 3 divided by x, let's see, x minus 1 times x plus 1 squared, integrated with respect to x.
00:21
I'm going to do this using a partial fraction decomposition.
00:25
So we're going to assume that this function f, right, f of x.
00:31
Which is x squared plus 2x plus 3 over x minus 1 so i wait for my pen to catch up x minus 1 times x plus 1 squared we're assuming this can be written in the form a over x minus 1 plus b over x plus 1 plus c over x plus 1 squared so let's take this function and right here, this equality, and let's cross multiply by all the terms in the denominator.
01:11
So what we'll have is x squared plus 2x plus 3 equals, and it'll be a times x plus 1 squared, plus b times x minus 1 times x plus 1, plus finally c times x minus 1.
01:34
All right.
01:35
So this is the equation.
01:36
We've got three variables here.
01:37
So if we set x equal to negative 1, for instance, on both sides, what do we get? so we'll get 1 minus 2 plus 3 on the left hand side.
01:48
On the right hand side, we will simply get negative 2c.
01:52
And so this indicates that c is equal to negative 1.
01:57
So that's important.
01:59
And then if we set x equal to positive 1, we can also make a further simplification.
02:06
On the right -hand side we have one plus two plus three which is six and then on the right -hand side we're going to have a times four and all the other terms are zero so this implies a is equal to three halves and then lastly if we substitute in these values of a and c into the equation and equate the terms like equate the coefficients of our polynomial so we'll have this on one side on the other side.
02:40
We'll have three halves times x plus one squared this term.
02:44
So when we square that term, we're going to get x squared plus 2x plus one.
02:50
And this is all multiplied by three halves.
02:53
And then we'll have b times x squared minus one.
03:00
Basically when we cross, when we multiply these terms out, we'll get an x squared minus one.
03:04
And then minus, it's like in our value for c.
03:08
Minus x plus one so the easiest way to find out what b is is let's uh distribute the three halves really quickly so we'll have three halves plus three x plus three halves here plus bx squared plus all these other terms so what we'll have is three halves the coefficient of x squared is going to be three halves plus b notice that so this comes out sorry my pen is failing me we'll have this term plus all these other other terms.
03:41
But this three halves plus b, the coefficient of x squared has to be equal on both sides.
03:45
So this has to equal one because the coefficient of x squared on the other side is equal to 1.
03:49
So this means b has to be equal to negative one half...