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Current time:0:00Total duration:3:52

AP.CALC:

FUN‑6 (EU)

, FUN‑6.E (LO)

, FUN‑6.E.1 (EK)

what we're going to do in this video is review the product rule that you probably learned a while ago and from that we're going to derive the formula for integration by parts which could really be viewed as the inverse product rule integration by parts so let's say that I start with some function that can be expressed as the product f of X it can be expressed as a product of two other functions f of X times G of X now let's take the derivative of this of this function let's apply the derivative operator right over here and this once again just review of the product rule is going to be the derivative of the first function times the second function so it's going to be F now I'm going to do that blue color it's going to be F that's not blue it's going to be F prime of x times G of x times that's not the same color times G of X G of X plus plus the first function times the derivative of the second plus the first function f of X times the derivative of the second derivative of the second this is all a review right over here the derivative of the first time the second function plus the first function times the derivative of the second function now let's take the antiderivative of both sides of this equation well if I take the antiderivative what I have here on the Left I get f of x times G of x times G of X we won't think about the constant for now we can ignore that for now and that's going to be equal to well what's the antiderivative of this which is going to be the antiderivative of F prime of X F prime of x times G of x times G of X DX DX plus the antiderivative of f of X f of X G prime of X G prime of X DX now what I want to do is solve I'm going to solve for this part right over here and to solve for that I just have to subtract this business I just have to subtract this business from both sides and then I am left with I'm life I subtract that from both sides I'm left with f of X times G of x times G of X minus this minus the antiderivative of F prime of X G of X G let me do that in pink color G of X G of X DX DX is equal to what I wanted to solve for is equal to the antiderivative of f of X G prime of X G prime of X DX DX and to make it a little bit clearer let me swap sides here so let me copy and paste this so let me copy and then paste it there you go and then let me copy and paste the other side so let me copy and paste it so I'm just switching the sides just to give it in a form that you might be that you might be more used to seeing in a calculus book so this tells us this is essentially the formula for integration by parts I will square it off you'll often see it squared off in a traditional textbook so I will do the same so this right over here tells us that if we have an integral or an antiderivative of the form f of X times the derivative of some other function we can apply this right over here and you might say what this doesn't seem that useful first I have to identify a function that's like this and then it's still I have an integral in it but what we will see in the next video is that this can actually simplify a whole bunch of things that you're trying to take the antiderivative of