MATH 128 - 1. Techniques of Integration (Chapter 7)...

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1.Techniques of Integration (Chapter 7)Integration by PartsTrigonometric IntegralsTrigonometric SubstitutionIntegration of Rational FunctionsDefinite vs. Indefinite IntegralsStrategy for IntegrationImproper IntegralsConvergence vs DivergenceComparison Theorem2.Applications of Integration (Chapter 8)Arc LengthArea of a Surface Revolution3.Differential Equations (Chapter 9)Modelling with DEDirection FieldsSeparable EquationsLinear EquationsApplications e.g. population growth4.Parametric Equations & Polar Coordinates (Chapter 10)Sketching curves defined by parametric equationsCalculus with parametric equationsPolar Coordinates5.Sequences & Series (Chapter 11)SequencesDivergence TestThe Integral Test and Estimates of SumsThe Comparison TestsAlternating SeriesAbsolute ConvergenceRatio TestRoot TestPower SeriesRepresentations of Functions as Power SeriesTaylor and Maclaurin SeriesTaylor’s Inequality
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Chapter 10 / Exercise 82
Applied Calculus
Berresford/Rockett
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Chapter 7: Techniques of IntegrationI. Integration by Parts???=?? − ???Example:?2??????Let?=?2,??= 2?????=??????,?=?????2??????=?2???? −????2? ??=?2???? −2????(?)??Evaluate this integral using integration by parts againLet?=?,??=????=??????,?=−???????????=−????? −−??????=−?????+??????=−?????+????+?Substitute back into the original equation:?2??????=?2???? −2(−?????+????+?)II. Trigonometric IntegralsStrategy for evaluating?????????????,? ≥0a) If m is even and n is odd:sin?????2?+1???=sin??(cos2?)???????=?????1− ???2????????Let?=????,??=??????=??1− ?2?−1??Expand, simplify, integrateExample:???2?cos3???=sin2?1− ???2?2??????=?21− ?22??=?62?4+?2
=?772?55+?33+?Substitute?=????=???7?72???5?5+???3?3+?b) If m is odd and n is even: Same as a) but we want everything as a function of cos instead of sin.Example:???3?cos2???=sin2?????(cos2?)??=(1cos2?)????(cos2?)??Let?=????,??=−??????Substitute, expand, simplify, integrate.c) If m and n are both odd, use either a) or b).d) If m and n are both even, use trig identities!Example:???4???=(sin2?)2??=1− ???2?22??=14(12 cos 2?+ cos22?)??Substitute the trig identitycos22?=12(1 +???4?)=1412???2?+121 +???4? ??=14322???2?+12???4? ??=14(32? −sin 2?+18???4?) +?We can use similar strategies for evaluatingtan??sec????using the identity???2?= 1 + tan2?a) If n is even:tan??sec2????=tan??(???2?)?−1sec2???=tan??(1 + tan2?)?−1sec2???Let?=????,??= sec2?=??(1 +?2)?−1??expand, simplify, integrateExample:tan6????4???=tan6?sec2?sec2???=tan6?(1 + tan2?) sec2???Let?=????,??= sec2?
=?61 +?2??=?77+?99+?=tan7?7+tan9?9+?b) If m is odd:tan2?+1?sec????=(tan2?)?sec?−1???????????=(sec2? −1)?????−1???????????Let?= sec?,??=??????????=(?21)???−1??Expand, simplify, integrateIII. Trig SubstitutionExpressionSubstitute?2− ?2?=????𝜃?2+?2?=????𝜃?2− ?2?=????𝜃Example:?2− ?2Let?=?sec𝜃=?2sec2𝜃 − ?2=?2(sec2𝜃 −1)=?2tan2𝜃=?tan𝜃With this in mind, let’s evaluate the integral???2−?2Let?=????𝜃,??=????𝜃???𝜃?𝜃=?sec𝜃tan𝜃?tan𝜃?𝜃=???𝜃?𝜃= ln???𝜃+???𝜃+?= ln |??+?2− ?2?| +?IV. Integration of Rational Functions? ?=?(?)?(?)f(x) is proper if degree of P < degree of Q, improper otherwise.

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Term
Winter
Professor
Zuberman
Tags
Math, Mathematical Series
We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
The document you are viewing contains questions related to this textbook.
Chapter 10 / Exercise 82
Applied Calculus
Berresford/Rockett
Expert Verified