4.1_C_lecture notes - AP BC Calc 4.1(contd contd...

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AP BC Calc – 4.1 (cont’d cont’d) Integration by Parts and Trigonometric Integrals Ferguson Notes In the first part of 4.1, the part addressing the method of Integration by Substitution, we found that the Substitution Rule was a response to differentiation by the Chain rule. That is, to integrate something that was differentiated by the Chain Rule, we must use Integration by Substitution. But what do we do if we’re faced with something that was differentiated using the Product (or Quotient) Rule? To integrate products (which includes quotients), we use a method called Integration by Parts . Note that the product rule says the following: . Therefore, . This equation can be rearranged as such: . Is that much better? No? Fine, then let and , which means and . By substitution, the equation becomes much easier to see and understand (and remember): Integration by parts is pretty much that….substitution. The hardest part, in my point of view is choosing the right functions for u and dv….. Ex. 1 – find . At this point, we can see that using the Substitution Rule is going to do us no good in evaluating this integral, so the next best method to try in integration by parts. We also knew that we’d need to integrate this by parts

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