Integration by PartsReview Product RuleManipulating the Product RuleSlide 4StrategyMaking the SplitStrategy HintTry ThisDouble TroubleGoing in CirclesApplicationAssignmentIntegration by PartsLesson 7.2Review Product Rule•Recall definition of derivative of the product of two functions•Now we will manipulate this to get[ ]( ) ( ) ( ) '( ) ( ) '( )xD f x g x f x g x g x f x� = � + �[ ]( ) '( ) ( ) ( ) ( ) '( )xf x g x D f x g x g x f x� = � - �Manipulating the Product Rule•Now take the integral of both sides•Which term above can be simplified?•This gives us[ ]( ) '( ) ( ) ( ) ( ) '( )xf x g x dx D f x g x dx g x f x dx� = � - �� � �( ) '( ) ( ) ( ) ( ) '( )f x g x dx f x g x g x f x dx� = � - �� �Integration by Parts•It is customary to write this using substitutionu = f(x) du = f '(x) dxv = g(x) dv = g'(x) dx( ) '( ) ( ) ( ) ( ) '( )f x g x dx f x g x g x f x dx� = � - �� �u dv u v v du= �-� �Strategy•Given an integral we split the integrand into two parts First part labeled uThe other labeled dv•Guidelines for making the split The dv always includes the dxThe dv must be integratable v du is easier to integrate than u dv Note: a certain amount of trial and error will happen in making this splitNote: a certain amount of trial and error will happen in making this splitxx e dx��u dv u v v du= �-� �Making the Split•A table to keep things organized is helpful•Decide what will be the u and the dv•This determines the du and the v•Now rewrite xx e dx��u dudv vxex dxdxexx xu v v du x e e dx�- � � -� �Strategy Hint•Trick is to select the correct function for u•A rule of thumb is the LIATE hierarchy ruleThe u should be first available fromLogarithmicInverse trigonometricAlgebraicTrigonometricExponentialTry This•Given•Choose a u and dv•Determinethe v and the du •Substitute the values, finish integration5lnx x dx�__________________u v v du�- ��u dudv vDouble Trouble•Sometimes the second integral must also be done by parts2sinx x dx�2cos 2 cosx x x x dx- + ��u dudv vu x2du 2x dxdv sin x v -cos xGoing in Circles•When we end up with the the same as we started with •Try•Should end up with•Add the integral to both sides, divide by 2 v du-�sinxe x dx�sin cos sin sinx x x xe x dx e x e x e x dx=- + -� �2 sin cos sinx x xe x dx e x e x=- +�Application•Consider the region bounded by y = cos x, y = 0, x = 0, and x = ½ π•What is the volume generated by rotatingthe region around the y-axis?What is the radius?What is the disk thickness?What are the limits?What is the radius?What is the disk thickness?What are the limits?Assignment•Lesson 7.2A•Page 439•Exercises 1 – 21 odd•Lesson 7.2B•Page 440•Exercises 35 – 41
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