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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Applied Calculus
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Chapter 10 / Exercise 82
Applied Calculus
Berresford/Rockett
Expert Verified

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