2015 AP Calculus BC 5d

0:00- [Voiceover] Let K equal six, so that F of X is equal to one over X-squared minus six X. Find the partial fraction decomposition for the function F. Find the integral of F of X D-X. And so let's first thinl about the partial fraction decomposition for the function F.

0:17So, F of X, I could rewrite it where I factor the denominator. If I factor out an X, I get X times X minus six. So, I can rewrite this as... And this is where I'm going to decompose it into partial fractions. A over X plus B over X minus six. If I actually had to add these two, I would try to find a common denominator.

0:48The best common denominator is just to take the product of these two expressions. So I could multiply the numerator and denomenator of this first time by X minus six. X minus six. And the numerator and denominator of the second term, I can multiply it by the denominator of the other one.

1:03So times X and times X. So that would give us, let's see if I distribute the A, it would give us A X minus six A plus B X over X times X minus six. If this looks completely foreign to you, I encourage you to watch the videos on Kahn Academy on partial fraction decomposition.

1:34So let's see if we can, so what we wanna do is solve for the A's and the B's. And so let's see, if we see that these have to add up to one over X times X minus six. So these have to add up. I'm just going back to this, these have to add up. The numerator. So it has to add up to one over X times X minus six.

2:01So this numerator has to add up to one. So what we can see is is that the X terms right over here must cancel out since we have no X terms here. We have A X plus B X must be equal to zero. Or you could say, well that means A plus B is equal to zero. So we took care of that term and that term.

2:24And then we know, that this must be the constant term that adds up to one or that is equal to one. And so we also know that negative six A is equal to one. Or A is equal to, divide both sides by negative six, negative one sixth. And if A is negative one sixth, well B is going to be the negative of that.

2:45And negative one sixth plus B. I'm just substituting A back into that equation. Need to be equal to zero. And so add one sixth to both sides, you get B is equal to one sixth. So I can decompose F of X. I can decompose F of X as being equal to A over X So that's negative one sixth over X.

3:10I just write it that way. I could write it as negative one over six X or something like that. But I'll just write it like this, just be clear that this was our A. Plus B, which is one sixth over X minus six. So that right over there, that's the partial fraction decomposition for our function F.

3:33And if I want to evaluate the integral. So the integral of F of X, the indefinite integral. Well, that's where this partial fractions decomposition is going to be valuable. That's going to be the indefinite integral of negative one sixth over X plus positive one sixth over X minus six, and then we have D X.

3:56Well what's the anti-derivative of this right over here? Well the anti-derivative of one over X is the natural log of the absolute value of X. And so we can just this is going to be negative one sixth times the natural log of the absolute value of X. Anti-derivative of this part.

4:12Then plus one sixth. You could do u-substitution but you could just say "Hey, look the derivative of this bottom part, X minus six." That's just one. So you could say that I have that derivative laying around. So I can just take the anti-derivative with respect to that.

4:29So that's going to be one sixth times the natural log of the absolute value of X minus six, and then I have plus C. Don't forget this is an indefinite integral over here. And then you're done. And you see that partial fraction decomposition was actually quite useful.

4:47So they were helping us how to figure out this. You didn't just have to have that insight that, okay how do I evaluate this anti-derivative. Well, the they're telling us to use partial fraction decomposition.