Integrating a product: integration by parts. Suppose we want to integrate Xsin(X). We can integrate each of these functions separately, but how do we integrate the product of them? The answer is we use a technique called "integration by parts". First I'll tell you the formula, then show you how it works with that particular example, and then explain where on earth the formula came from! Here's the method. We have a product of two functions and we want to integrate this product. The first step is, we have to choose one of them to be integrated and one to be differentiated. (The result will be that the whole product gets integrated, but to get there we actually have to differentiate one of them. It seems strange but you'll see why later.) So we choose one to integrate and the other to differentiate. To remind us which one we're going to integrate we'll write it as the derivative of something, let's call it dv/dX, so when we integrate i...
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