yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

2015 AP Calculus BC 5d | AP Calculus BC solved exams | AP Calculus BC | Khan Academy


3m read
·Nov 11, 2024

Let k equals 6, so that f of x is equal to 1 over x squared minus 6x. Find the partial fraction decomposition for the function f, find the integral of f of x dx.

And so, let's first think about the partial fraction decomposition for the function f. So, f of x, I could rewrite it where I factor the denominator. If I factor out an x, I get x times x minus 6, and 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 6.

And if I actually had to add these two, I would try to find a common denominator. The best common denominator is just to take the product of these two expressions. So, I can multiply the numerator and the denominator of this first term by x minus 6, and then the numerator and denominator of the second term I can multiply by the denominator of the other one, so times x and times x.

And so that would give us— let’s see, if I distribute the a, it would give us a x minus 6a plus bx plus b over x times x minus 6. If this looks completely foreign to you, I encourage you to watch the videos on Khan Academy on partial fraction decomposition.

All right, so let's see if we can— so what we want to do is we want to solve for the a's and we want to solve for the b's. And so let’s see if we see that these have to add up to 1 over x times x minus 6. So, these have to add up— I’m just going back to this— these have to add up to the numerator, so it has to add up to 1 over x times x minus 6.

And so, this numerator has to add up to 1. So, what we can see is that the x terms right over here must cancel out, since we have no x terms here. So, we have a x plus b x must be equal to zero, or you could say, well that means that a plus b is equal to zero. So, we took care of that term and that term, and 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 6a is equal to one, or a is equal to— divide both sides by negative 6— negative 1/6. And then if a is negative 1/6, well b is going to be the negative of that. We have negative 1/6 plus b— I’m just substituting a back into that equation— need to be equal to 0. And so, add 1/6 to both sides, you get b is equal to 1/6.

So, I can decompose f of x. I can decompose f of x as being equal to a over x, so that’s negative 1/6 over x. I just write it that way. I could write it as negative 1 over 6x or something like that, but I’ll just write it like this just to be clear that this was our a plus b, which is 1/6 over x minus 6 over x minus 6.

So that right over there, that’s the partial fraction decomposition for our function f. And if I want to evaluate the integral— so the integral of f of x, the indefinite integral— well, that’s where this partial fraction decomposition is going to be valuable. That’s going to be the indefinite integral of negative 1/6 over x plus positive 1/6 over x minus 6, and then we have dx.

Well, what’s the anti-derivative of this right over here? Well, the anti-derivative of 1 over x is the natural log of the absolute value of x. And so, we can just say this is going to be negative 1/6 times the natural log of the absolute value of x. That’s the anti-derivative of this part, and then plus 1/6— plus 1/6.

You could do u substitution, but you could just say, hey look, the derivative of this bottom part x minus 6, that’s just 1. And so you could say that, okay, I have that derivative laying around and so this is— so I can just take the anti-derivative with respect to that. And so that’s going to be 1/6 times the natural log of the absolute value of x minus 6.

And then I have— 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 that partial fraction decomposition was actually quite useful, so they were helping us how to figure out this. You didn’t just have to have that insight—they're like, okay, how do I evaluate this anti-derivative? Well, they’re telling us to use partial fraction decomposition.

More Articles

View All
15 of the Worst Life Decisions Anyone Can Make
What is rock bottom, really? Perhaps it’s sitting outside alone in the dark. You’re broke, no friends or family to call, feeling mentally and physically sick. Your addiction, calling your name with no end in sight, sounds pretty rock bottom. But how does …
The Last Days of the Romanovs | National Geographic
I think it’s a big tragedy, big tragedy for the country and for the world. For 300 years, the Romanovs ruled Russia as czars—loved, feared, revered, respected. But all too often, those who fly highest fall furthest. World War One brought Russia to revolut…
STOP SPENDING MONEY (Major Changes To ALL Credit Cards)
What’s up Grandma! It’s guys here, so no need to worry about rising interest rates, high inflation, heated consumer spending, or Microsoft’s new AI exposing personal information out of vengeance. Because instead, the latest threat to our economy is said t…
Crisis | Vocabulary | Khan Academy
Wordsmiths, we’re in it now, you and I. The situation has become very serious. You might even say it’s a crisis. Yes, crisis is the word we’re going to be looking at in this video. Crisis, it’s a noun. It means a tipping point, a very dangerous period or …
Grammatical person and pronouns | The parts of speech | Grammar | Khan Academy
Serious question, Grimian: What’s the difference between me and you? Uh, well, in order to get… I mean, I don’t mean that, you know, in a snarky way. I mean that in like a conceptual way. What’s the difference? Uh, in terms of these two pronouns, what’s s…
How to Create Luck - Dalton Caldwell, Y Combinator Partner
I’m Dalton. I’m a partner at Y Combinator. I was the founder of a company called imeem in 2003 and a company called mixed-media labs in 2010. I’m working at YC since 2013. Okay, how do you create luck? The way to create luck is to move much faster than e…