00:01
Let's evaluate the given integral.
00:03
The first thing to observe here is that the numerator has larger degree than the denominator.
00:09
So we should do polynomial division to simplify before we even consider partial fractions.
00:16
So let's do the division here.
00:18
X to the 4 divided by x squared.
00:21
That's just x squared.
00:24
Let's go ahead and multiply that out.
00:29
Then we subtract.
00:32
And we're left over with x plus 2 because we have cancellation here.
00:37
So this tells us that we can rewrite the integral as x plus 2, oops, excuse me, x squared, this is the quotient, and then we have our remainder x plus 2 over the original denominator, x squared plus 9.
01:01
So this is a simpler looking integral than the original.
01:05
There's really no need for partial fractions here because this term over here already is a partial fraction.
01:13
We have a quadratic in the denominator that does not factor, and then you have your ax plus b in the numerator.
01:20
So really, we're ready to integrate.
01:24
Let's break this into three integrals.
01:27
And then we could split up this fraction.
01:30
This is just x over x squared plus 9 plus 2 over x squared plus 9.
01:38
So let's go ahead and integrate each of those.
01:47
And then for the last one, let's go ahead and pull out that 2.
01:55
And now we have three integrals.
02:01
So the first one, just use the power rule...