Basic Integration RulesFitting Integrals to Basic RulesTry It OutThe Log Rule in DisguiseThe Power Rule in DisguiseSlide 6Disguises with Trig IdentitiesAssignmentIntegration by PartsReview Product RuleManipulating the Product RuleSlide 12StrategyMaking the SplitStrategy HintTry ThisDouble TroubleGoing in CirclesApplicationSlide 20Trigonometric IntegralsRecall Basic IdentitiesIntegral of sinn x, n OddSlide 24Integral of sinn x, n EvenCombinations of sin, cosSlide 27Combinations of tanm, secnSlide 29Integrals of Even Powers of sec, cscWallis's FormulasSlide 32AssignmentTrigonometric SubstitutionNew Patterns for the IntegrandExampleFinishing UpKnowing Which SubstitutionTry It!!Keep Going!Slide 41Special Integration FormulasSlide 43Partial FractionsPartial Fraction DecompositionSlide 46The ProcessSlide 48Slide 49Slide 50A VariationSlide 52What IfGotta Try ItSlide 55Even More ExcitingCombine the MethodsTry It This TimePartial Fractions for IntegrationWhy Are We Doing This?Slide 61Integration by TablesTables of IntegralsGeneral Table ClassificationsFinding the Right FormReduction FormulasSlide 67Slide 68Indeterminate Forms and L’Hopital’s RuleProblemExample of the ProblemL’Hopital’s RuleSlide 73Slide 74Slide 75HintsSlide 77Improper IntegralsSlide 79To Infinity and BeyondSlide 81To Limit Or Not to LimitTo Converge Or NotImproper Integral to -When f(x) Unbounded at x = cUsing L'Hopital's RuleSlide 87Slide 88Basic IntegrationRulesLesson 8.1Fitting Integrals to Basic Rules•Consider these similar integrals•Which one uses …•The log rule•The arctangent rule•The rewrite with long division principle22 2 25 5 54 4 4x xdx dx dxx x x+ + +� � �Try It Out•Decide which principle to apply …21xdxx +�( )222 1 4dtt - +�The Log Rule in Disguise•Consider•The quotient suggests possible Log Rule, but the _________ is not present•We can manipulate this to make the Log Rule apply •Add and subtract ex in the numerator11xdxe+�11x xxe edxe+ -=+�The Power Rule in Disguise•Here's another integral that doesn't seem to fit the basic options•What are the options for u ?•Best choice is( ) ( )cot ln sinx x dx�� �� ��________ __________________u du= =The Power Rule in Disguise•Thenbecomes and _____________applies •Note review of basic integration rules pg 520•Note procedures for fitting integrands to basic rules, pg 521( ) ( )cot ln sinx x dx�� �� ��u du�Disguises with Trig Identities•What rules might this fit?•Note that tan2 u is ____________________•However sec2u is on the list•This suggests one of the _____________________identities and we have2tan 2x dx�Assignment•Lesson 8.1•Page 522•Exercises 1 – 49 EOOIntegration by PartsLesson 8.2Review 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� = - �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� =�Integration by Parts•It is customary to write this using substitution•u = f(x) du = ____________•v = g(x) _________ = 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 u•The other labeled dv•Guidelines for making the split• The dv always includes the _______•The ______ must be integratable •v du is ___________________________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 _____ and the _____•This determines the du and the v•Now rewrite xx e dx��u dudv vx 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 from•L___________________•Inverse trigonometric•A___________•Trigonometric•E________________Try 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_____________v du-�sinxe 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 8.2A•Page 531•Exercises 1 – 35 odd•Lesson 8.2B•Page 532•Exercises 47 – 57, 99 – 105 oddTrigonometric IntegralsLesson 8.3Recall Basic Identities•Pythagorean Identities•Half-Angle Formulas2 22 22 2sin cos 1tan 1 sec1 cot cscq qq qq q+ =+ =+ =221 cos 2sin21 cos 2cos2qqqq-=+=These will be used to integrate powers of sin and cosThese will be used to integrate powers of sin and cosIntegral of sinn x, n Odd•Split into product of an __________________•Make the even power a power of sin2 x•Use the Pythagorean identity•Let u = cos x, du = -sin x dx5 4sin sin sinx dx x x dx= �� �( )24 2sin sin sin sinx x dx x x dx� =� �( )22sin sin ___________________x x dx =�( )22 2 41 1 2 ...u du u u du- - =- - + =� �Integral of sinn x, n Odd•Integrate and un-substitute•Similar strategy with cosn x, n odd2 4 3 52 11 23 5__________________________u u du u u u C- - + =- + - +=�Integral of sinn x, n Even•Use half-angle formulas•Try Change to power of ________•Expand the binomial, then integrate21 cos 2sin2qq-=4cos 5x dx�( )( )2221cos 5 1 cos102x dx x dx� �= +� �� �� �Combinations of sin, cos•General form•If either n or m is odd, use techniques as before•Split the _____ power into an