How does integration by parts work?
In this video we talk about why the integration by parts formula actually works. What we conclude is that there are two reasons.
0:36 Where does integration by parts come from? // First, the integration by parts formula is a result of the product rule formula for derivatives. In a lot of ways, this makes sense. After all, the product rule formula is what lets us find the derivative of the product of two functions. So, if we want to find the integral of the product of two functions, why wouldn’t we try integrating the product rule? In fact, that’s exactly how we get to the integration by parts formula. We start with the product rule, and we integrate both sides. Through some fancy rearranging, we end up with the integration by parts formula. And so we then have the product rule formula for taking the derivative of a product, and we have the related integration by parts formula for taking the integral of a product.
1:09 Why does the integration by parts formula work? // Second, the integration by parts formula works because it takes an integrand that we CAN’T integrate, and turns it into an integrand that we CAN integrate. And that’s the same as any other method of integration, like substitution, partial fractions, or trig substitution, to name a few. They all reformat the integrand so that it’s easier for us to integrate. If we look at the integration by parts formula, we can see that it changes the integral of u times dv to the integral of v times du. So if the integral of v du is easier for us to integrate than the integral of u dv, then the integration by parts formula is going to be a huge help to us!
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