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

2. BMayer@ChabotCollege.edu 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-22 5.1

3. BMayer@ChabotCollege.edu 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

4. BMayer@ChabotCollege.edu 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

5. BMayer@ChabotCollege.edu 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

6. BMayer@ChabotCollege.edu 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 →

7. BMayer@ChabotCollege.edu 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.

8. BMayer@ChabotCollege.edu 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 →

9. BMayer@ChabotCollege.edu 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.

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

11. BMayer@ChabotCollege.edu 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?

12. BMayer@ChabotCollege.edu 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

13. BMayer@ChabotCollege.edu 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:

14. BMayer@ChabotCollege.edu 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)

15. BMayer@ChabotCollege.edu 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

16. BMayer@ChabotCollege.edu 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

17. BMayer@ChabotCollege.edu 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:

18. BMayer@ChabotCollege.edu 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:

19. BMayer@ChabotCollege.edu 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

20. BMayer@ChabotCollege.edu 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 →

21. BMayer@ChabotCollege.edu 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)

22. BMayer@ChabotCollege.edu 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

23. BMayer@ChabotCollege.edu 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

24. BMayer@ChabotCollege.edu 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:

25. BMayer@ChabotCollege.edu 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

26. BMayer@ChabotCollege.edu 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.

27. BMayer@ChabotCollege.edu 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

28. BMayer@ChabotCollege.edu 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:

29. BMayer@ChabotCollege.edu 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:

30. BMayer@ChabotCollege.edu 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 + 0.026A 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:

31. BMayer@ChabotCollege.edu 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

32. BMayer@ChabotCollege.edu 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

33. BMayer@ChabotCollege.edu 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

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

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

36. BMayer@ChabotCollege.edu 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

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

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

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

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

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

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

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

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

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

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

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