MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §5.2 Integration By Substitution

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §5.1 → AntiDerivatives  Any QUESTIONS About HomeWork §5.1 → HW

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 3 Bruce Mayer, PE Chabot College Mathematics §5.2 Learning Goals  Use the method of substitution to find indefinite integrals  Solve initial-value and boundary-value problems using substitution  Explore a price-adjustment model in economics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 4 Bruce Mayer, PE Chabot College Mathematics Recall: Fcn Integration Rules 1.Constant Rule: for any constant, k 2.Power Rule: for any n≠−1 3.Logarithmic Rule: for any x≠0 4.Exponential Rule: for any constant, k

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 5 Bruce Mayer, PE Chabot College Mathematics Recall: Integration Algebra Rules 1.Constant Multiple Rule: For any constant, a 2.The Sum or Difference Rule: This often called the Term-by-Term Rule

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 6 Bruce Mayer, PE Chabot College Mathematics Integration 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, x  In the AntiDerivative expression ∫f(x)dx substitutions must be made: Within the Integrand For dx  Along Lines →

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 7 Bruce Mayer, PE Chabot College Mathematics Investigate Substitution  Compute the family of AntiDerivatives given by a.by expanding (multiplying out) and using rules of integration from Section 5.1 b.by writing the integrand in the form u 2 and guessing at an antiderivative.

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 8 Bruce Mayer, PE Chabot College Mathematics Investigate Substitution  SOLUTION a:  “Expand the BiNomial” by “FOIL” Multiplication  SOLUTION b:  Let:  Sub u into Expression →

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 9 Bruce Mayer, PE Chabot College Mathematics Investigate Substitution  Examine the “substituted” expression to find the Integrand stated in terms of u Integrating factor (dx) stated in terms of x  The 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.

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 10 Bruce Mayer, PE Chabot College Mathematics Investigate Substitution  Continuing

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 11 Bruce Mayer, PE Chabot College Mathematics Investigate Substitution  The integral in part (b) (which is speculative) agrees with the integral calculated in part (a) (using established techniques) when  By Correspondence observe that ?=⅓ This Begs the Question: is there some systematic, a-priori, method to determine the value of the question-mark?

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 12 Bruce Mayer, PE Chabot College Mathematics SubOut Integrating Factor, dx  Let the single value, u, represent an algebraic expression in x, say:  Then take the derivative of both sides  Then Isolate dx

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 13 Bruce Mayer, PE Chabot College Mathematics SubOut Integrating Factor, dx  Then the Isolated dx:  Thus the SubStitution Components  Consider the previous example  Let:  Then after subbing:

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 14 Bruce Mayer, PE Chabot College Mathematics SubOut Integrating Factor, dx  Now Use Derivation to Find dx in terms of du →  Multiply both sides by dx/3 to isolate dx  Now SubOut Integrating Factor, dx  Now can easily AntiDerivate (Integrate)

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 15 Bruce Mayer, PE Chabot College Mathematics SubOut Integrating Factor, dx  Integrating  Recall:  BackSub u=3x+1 into integration result  Expanding the BiNomial find

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 16 Bruce Mayer, PE Chabot College Mathematics SubOut Integrating Factor, dx  Then  The Same Result as Expanding First then Integrating Term-by-Term Using the Sum Rule

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 17 Bruce Mayer, PE Chabot College Mathematics GamePlan: Integ by Substitution 1.Choose a (clever) substitution, u = u(x), that “simplifies” the Integrand, f(x) 2.Find the Integrating Factor, dx, in terms of x and du by:

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 18 Bruce Mayer, PE Chabot College Mathematics GamePlan: Integ by Substitution 3.After finding dx = r(h(u), du) Sub Out the Integrand and Integrating Factor to arrive at an equivalent Integral of the form: 4.Evaluate the transformed integral by finding the AntiDerivative H(u) for h(u) 5.BackSub u = u(x) into H(u) to eliminate u in favor of x to obtain the x-Result:

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example: Substitution with e  Find  SOLUTION:  First, note that none of the rules from the Previous lecture on §5.1 will immediately resolve this integral  Need to choose a substitution that yields a simpler integrand with which to work Perhaps if the radicand were simpler, the §5.1 rules might apply

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example: Substitution with e  Try Letting:  Take d/dx of Both Sides  Solving for dx:  Now from u-Definition:  Then dx →

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example: Substitution with e  Now Sub Out in original AntiDerivative:  This process yields  This works out VERY Well  Now can BackSub for u(x)

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example: 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

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example: Sub Rational Expression  Find  SOLUTION:  Try:  Taking du/dx find  This produces

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example: Sub Rational Expression  Solving  Thus the Answer  An Alternative u:

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example: Sub Rational Expression  SubOut x using:  Find  Then The Same Result as before

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  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.

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  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·A –Where A is the current Amount in the 403b  Then the yearly Rate of Change for the Amount in the 403b account

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  DE Model for Annuities  This DE is Variable Separable  Affecting the Separation and Integrating  Find the AntiDerivative by Substitution  Let:  Then:

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example  DE Model for Annuities  SubOut A in favor of u:  Integrating:

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example  DE Model for Annuities  Note that u = $40k A is always positive, so the ABS-bars can be dispensed with  Now BackSub  Solve for A(t) by raising e to the power of both sides Find the General (Includes C) solution:

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 31 Bruce Mayer, PE Chabot College Mathematics Example  DE Model for Annuities  Use the KNOWN data that at year-Zero there is NO money in the 403b; i.e.; (t 0,A 0 ) = (0,A(0)) = (0,0)  Sub (0,0) into the General Soln to find C  Or  Thus the particular soln

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example  DE Model for Annuities  Using the Log property  Find  Factoring Out the 40  Then at 10 years the 403b Amount

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 33 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §5.2 P61 → Retirement Income vs. Outcome P66 → Price Sensitivity to Supply & Demand

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 34 Bruce Mayer, PE Chabot College Mathematics All Done for Today Substitution City

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 35 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 36 Bruce Mayer, PE Chabot College Mathematics ConCavity Sign Chart abc −−−−−−++++++−−−−−− x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 37 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 38 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 39 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 40 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 41 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 42 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 43 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 44 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 45 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 46 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-23_sec_5-2_Integration_Substitution.pptx 47 Bruce Mayer, PE Chabot College Mathematics