4.1_C_lecture notes - AP BC Calc 4.1(contd contd Integration by Parts and Trigonometric Integrals Ferguson Notes In the first part of 4.1 the part

# 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: ( ) ( ) ( ) ( ) ( ) ( ) f x g x f x g x f x g x . Therefore, ( ) ( ) ( ) ( ) ( ) ( ) f x g x f x g x dx f x g x C . This equation can be rearranged as such: ( ) ( ) ( ) ( ) ( ) ( ) f x g x dx f x g x f x g x dx . Is that much better? No? Fine, then let ( ) f x u and ( ) g x v , which means ( ) du f x dx and ( ) dv g x dx . By substitution, the equation becomes much easier to see and understand (and remember): udv uv vdu 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 sin x xdx . 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 because it is a product of two functions (not always a reliable method of remembering, though). If we let u x and sin dv x , then du dx and cos v x   . So by utilizing our integration by parts method, sin cos cos cos cos cos sin x xdx x x xdx x x xdx x x x C       . Done. “Well that was easy.” Ex. 2 find ln xdx . Let ln u x and dv dx , then 1 du x and v x . 1 ln ln ln ln xdx x x x dx x x dx x x x C x . Wow these integrals are going quickly!