Integration by parts
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Integration by parts intro
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, 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 function, let's apply the derivative operator right over here. And this, once again, just a review of the product rule. It's going to be the derivative of the first function times the second function. So it's going to be f-- no, 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 plus the first function times the derivative of the second, plus the first function, f of x, times the derivative of the second. This is all a review right over here. The derivative of the first times 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 of what I have here on the left, I get f 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? This is going to be the antiderivative of f prime of x times g of x dx plus the antiderivative of f of x g prime of x dx. Now, what I want to do is 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 if I subtract that from both sides, I'm left with f of x times g of x minus this, minus the antiderivative of f prime of x g of x-- let me do that in a pink color-- g of x dx is equal to what I wanted to solve for, is equal to the antiderivative of f of x g prime of x 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 more used to seeing in a calculus book. So 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, well this doesn't seem that useful. First I have to identify a function that's like this. And then still I have an integral in it. But what we'll 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.