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
How to Find the Right Mentor | Ask Mr. Wonderful Shark Tank's Kevin O'Leary
So my question is: how can a 22-year-old make himself useful or stand out to a business person that can perhaps take a risk to pull me along and teach me what are the skills and things you would need to see in a candidate to even consider teaching him? He…
The Most Successful Shark Tank Deals and Products | Kirk Minihane
[Music] Good, so you’re live now on the Kirkman the Hand show. Uh, nice to meet you. Normally, normally, I said this earlier, I am podcast wonderful, but I’m gonna hand that over to you today out of respect. I think you’ve earned that; you’ve earned that …
9 Japanese Philosophies to Become Self-Disciplined and Stop Procrastinating
Have you ever struggled with procrastination or finding the motivation to get things done, feeling like you’re not living up to your own expectations? I won’t lie. I struggle with procrastination a lot, and it’s a challenge, especially when I have importa…
What Does Earth Look Like?
Hey, Vsauce. Michael here. This point of light in the sky is Earth as seen from the surface of Mars. And this is Earth as seen from Saturn. Here’s an image taken only 45,000 kilometres away, the famous Blue Marble. But what does Earth really look like? We…
Estimating actual COVID 19 cases (novel corona virus infections) in an area based on deaths
The goal of this video is to help us all estimate the actual new COVID-19 cases per day in your area, and it’s based on analysis by Thomas Pueyo. He wrote an incredible blog post on Medium; this is the link, and I’ll also include it in the description bel…
Disability and long term care insurance | Insurance | Financial literacy | Khan Academy
Talking about insurance is never a lot of fun because you’re talking about thinking, or you’re thinking about things that most of us don’t want to have to deal with in life. I’m going to continue that trend by talking about two new scenarios of insurance,…