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!
Music by Joakim Karud: http://soundcloud.com/joakimkarud
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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student—from basic middle school classes to advanced college calculus—figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: http://www.kristakingmath.com
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