# Bruce Mayer, PE

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Bruce Mayer, PELicensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

Chabot Mathematics

§5.2 IntegrationBy Substitution

Slide2Review §

Any QUESTIONS About§5.1 → AntiDerivativesAny QUESTIONS About HomeWork§5.1 → HW-22

5.1

Slide3§5.2 Learning Goals

Use the method of substitution to find indefinite integralsSolve initial-value and boundary-value problems using substitutionExplore a price-adjustment model in economics

Slide4Recall: Fcn Integration Rules

Constant Rule: for any constant, kPower Rule:for any n≠−1Logarithmic Rule:for any x≠0Exponential Rule:for any constant, k

Slide5Recall: Integration Algebra Rules

Constant Multiple Rule: For any constant, aThe Sum or Difference Rule:This often called the Term-by-Term Rule

Slide6Integration by Substitution

Sometimes it is MUCH EASIER to find an AntiDerivative by allowing a new variable, say u, to stand for an entire expression in the original variable, xIn the AntiDerivative expression ∫f(x)dx substitutions must be made:Within the IntegrandFor dxAlong Lines →

Slide7Investigate Substitution

Compute the family of AntiDerivatives given byby expanding (multiplying out) and using rules of integration from Section 5.1by writing the integrand in the form u2and guessing at an antiderivative.

Slide8Investigate Substitution

SOLUTION a:“Expand the BiNomial” by “FOIL” MultiplicationSOLUTION b:Let:Sub u into Expression →

Slide9Investigate Substitution

Examine the “substituted” expression to find theIntegrand stated in terms of uIntegrating factor (dx) stated in terms of xThe Integrand↔IntegratingFactor MisMatch does Not Permit the AntiDerivation to move forward.Let’s persevere, with the understanding is something missing by flagging that with a (well-placed) question mark.

Slide10Investigate Substitution

Continuing

Slide11Investigate Substitution

The integral in part (b) (which is speculative) agrees with the integral calculated in part (a) (using established techniques) whenBy Correspondence observe that ?=⅓This Begs the Question: is there some systematic, a-priori, method to determine the value of the question-mark?

Slide12SubOut Integrating Factor, dx

Let the single value, u, represent an algebraic expression in x, say:Then take thederivative of bothsidesThen Isolate dx

Slide13SubOut Integrating Factor, dx

Then the Isolated dx:Thus the SubStitution ComponentsConsider the previous exampleLet: Then after subbing:

Slide14SubOut Integrating Factor, dx

Now Use Derivation to Find dx in terms of du →Multiply both sides by dx/3 to isolate dxNow SubOut Integrating Factor, dxNow can easily AntiDerivate (Integrate)

Slide15SubOut Integrating Factor, dx

IntegratingRecall:BackSub u=3x+1 into integration resultExpanding the BiNomial find

Slide16SubOut Integrating Factor, dx

ThenThe Same Result as Expanding First then Integrating Term-by-Term Using the Sum Rule

Slide17GamePlan: Integ by Substitution

Choose a (clever) substitution, u = u(x), that “simplifies” the Integrand, f(x)Find the Integrating Factor, dx, in terms of x and du by:

Slide18GamePlan: Integ by Substitution

After finding dx = r(h(u), du) Sub Out the Integrand and Integrating Factor to arrive at an equivalent Integral of the form:Evaluate the transformed integral by finding the AntiDerivative H(u) for h(u)BackSub u = u(x) into H(u) to eliminate u in favor of x to obtain the x-Result:

Slide19Example: Substitution with e

FindSOLUTION:First, note that none of the rules from the Previous lecture on §5.1 will immediately resolve this integralNeed to choose a substitution that yields a simpler integrand with which to workPerhaps if the radicand were simpler, the §5.1 rules might apply

Slide20Example: Substitution with e

Try Letting:Take d/dx of Both SidesSolving for dx: Now from u-Definition:Thendx →

Slide21Example: Substitution with e

Now Sub Out in original AntiDerivative:This process yieldsThis works out VERY WellNow can BackSub for u(x)

Slide22Example: Substitution with e

Using u(x) = e−x+7:Thus the Final Result:This Result can be verified by taking the derivative dZ/dx which should yield the original integrand

Slide23Example: Sub Rational Expression

FindSOLUTION:Try:Taking du/dx find This produces

Slide24Example: Sub Rational Expression

SolvingThus the AnswerAn Alternative u:

Slide25Example: Sub Rational Expression

SubOut x using:FindThenThe Same Result as before

Slide26Example DE Model for Annuities

Li Mei is a

Government Worker

who has an annuity referred to as a

403b

. She deposits money continuously into the 403b at a rate of

$40,000 per year

, and it earns

2.6% annual interest

.

Write a differential equation modeling the growth rate of the net worth of the annuity, solve it, and determine how much the annuity is worth at the end of 10 years.

Slide27Example DE Model for Annuities

SOLUTION:TRANSLATE: The 403b has two ways in which it grows yearly: The annual Deposit by Li Mei = $40k The annual interest accrued = 0.026·AWhere A is the current Amount in the 403bThen the yearly Rate of Change for the Amount in the 403b account

Slide28Example DE Model for Annuities

This DE is Variable SeparableAffecting the Separation and IntegratingFind the AntiDerivative by SubstitutionLet:Then:

Slide29Example DE Model for Annuities

SubOut A in favor of u:Integrating:

Slide30Example DE Model for Annuities

Note that u = $40k + 0.026A is always positive, so the ABS-bars can be dispensed withNow BackSubSolve for A(t) by raising e to the power of both sidesFind the General(Includes C) solution:

Slide31Example DE Model for Annuities

Use the KNOWN data that at year-Zero there is NO money in the 403b; i.e.; (t0,A0) = (0,A(0)) = (0,0)Sub (0,0) into the General Soln to find COrThus the particular soln

Slide32Example DE Model for Annuities

Using the Log propertyFindFactoring Out the 40Then at 10 years the 403b Amount

Slide33WhiteBoard Work

Problems From §5.2P61 → Retirement Income vs. OutcomeP66 → Price Sensitivity to Supply & Demand

Slide34All Done for Today

Substitution

City

Slide35Bruce Mayer, PELicensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

Chabot Mathematics

Appendix

–

Do On

Wht

/

Blk

Borad

Slide36ConCavity Sign Chart

a

b

c

−−−−−−

++++++

−−−−−−

++++++

x

ConCavity

Form

d

2

f

/

dx

2

Sign

Critical (Break)

Points

Inflection

NOInflection

Inflection

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## Bruce Mayer, PE

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