Differential Equations - Review and Introduction - Quick Guides

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Study GuideDifferential Equations–Review and Introduction1. Integration1.1 What Is Integration?Integration is one of the most important ideas in calculus.It helps us answer two big questions:1.What function has a given derivative?2.What is the area under a curve?Because of this, integration comes intwo main types:•Indefinite Integration•Definite IntegrationIndefinite Integration (Antidifferentiation)Indefinite integration is thereverse of differentiation.This means:•If differentiation finds thederivative,•Indefinite integration finds theoriginal function.Suppose you are given a function(f(x)).Indefinite integration asks:“What function(F(x))has a derivative equal to(f(x))?”That function(F(x))is called anantiderivative.NotationWe write an indefinite integral as:

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Study Guide•Theintegral sign(∫)means integrate•(f(x))is the function being integrated•(dx)shows that(x)is the variable of integrationSince many functions can have the same derivative, we always add aconstant of integration,written as+ C.Definite Integration (Area Under a Curve)Definite integration has ageometric meaning.It is used tocalculate the areabetween:•The curve(y = f(x))•Thex-axis•Two vertical lines(x = a)and(x = b)In simple words, it tells ushow much area lies under the curvebetween two points.Notation•(a)is thelower limit•(b)is theupper limit•The result is anumber, not a function

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Study GuideFigure 11.2 How Indefinite and Definite Integrals Are ConnectedAt first glance, these two types of integration seem very different:•Indefinite integrationgives afamily of functions•Definite integrationgives asingle numerical valueSo why are they both called “integration”?The connection comes from one of the most important results in calculus.1.3 The Fundamental Theorem of CalculusTheFundamental Theorem of Calculus (FTC)explains the deep link between:•Differentiation•IntegrationIt hastwo parts.1.4 Fundamental Theorem of Calculus–Part IIf a function(f(t))is continuous, then:

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Study Guide1.5 Fundamental Theorem of Calculus–Part IIIf(f(x))is continuous and has an antiderivative(F(x)), then:This shows thatarea under a curve can be found using antiderivatives.Example: Indefinite IntegrationEvaluateSolutionWe integrate each term separately:Final AnswerHere,Cis the constant of integration.1.6 Common Differentiation and Integration FormulasTo solve integrals efficiently, it’s important to remember standard formulas.These formulas show:•A function•Its derivative•Its corresponding integral

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Study GuideThey include:•Powers•Exponential functions•Logarithmic functions•Trigonometric functions•Inverse trigonometric functionsThis table is best placed at the end so students can:•Refer to it while solving problems•Use it as a quick revision sheet

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Study GuideSummary•Indefinite integralsfind antiderivatives•Definite integralscalculate area•TheFundamental Theorem of Calculusconnects both ideas•Integration is a powerful tool used throughout mathematics, physics, and engineering2. Techniques of Indefinite IntegrationWhen basic integration formulas are not enough, we usespecial techniquesto evaluate integrals.In this chapter, we focus on two important methods:1.Integration by Substitution2.Integration by PartsLet’s learn them one at a time with clear examples.2.1 Integration by SubstitutionIntegration by substitution is themost commonly used technique.It is especially helpful when the integrand contains acomplicated expression.Main Idea•Replace a complicated part of the integrand with asingle variable, usually(u).•If the differential(du)also appears (or can be made to appear), the integral becomes easy.If after substitution the integral simplifies nicely, your choice of(u)is correct.Example 1:Evaluate

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Study GuideStep 1: Choose a substitutionLetStep 2: DifferentiateStep 3: Rewrite the integralStep 4: IntegrateExample 2:IntegrateLetThen:Substitute back:

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Study GuideExample 3:EvaluateRewrite:LetThen:Example 4:EvaluateLetThen:Example 5:DetermineLet

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Study GuideThen:2.2 Integration by PartsSometimes substitution doesnotsimplify the integral.This happens when the integrand is aproduct of two functions.For such cases, we useintegration by parts.Formula for Integration by PartsFrom the product rule:Integrating both sides:Example 6:IntegrateHere, substitution does not help, so we use integration by parts.Choose:

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Study GuideApply the formula:Example 7:IntegrateLet:Then:Example 8:EvaluateLet:Then:

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