Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §5.2 IntegrationBy Substitution Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 5.1 Review § • Any QUESTIONS About • §5.1 → AntiDerivatives • Any QUESTIONS About HomeWork • §5.1 → HW-22

3. §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. Recall: Fcn Integration Rules • Constant Rule: for any constant, k • Power Rule:for any n≠−1 • Logarithmic Rule:for any x≠0 • Exponential Rule:for any constant, k

5. Recall: Integration Algebra Rules • Constant Multiple Rule: For any constant, a • The Sum or Difference Rule: • This often called the Term-by-Term Rule

6. 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. Investigate Substitution • Compute the family of AntiDerivativesgiven by • by expanding (multiplying out) and using rules of integration from Section 5.1 • by writing the integrand in the form u2and guessing at an antiderivative.

8. Investigate Substitution • SOLUTION a: • “Expand the BiNomial” by “FOIL” Multiplication • SOLUTION b: • Let: • Sub u into Expression →

9. 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↔IntegratingFactorMisMatch does Not Permit the AntiDerivation to move forward. • Let’s persevere, with the understanding is something missing byflagging that with a (well-placed) question mark.

10. Investigate Substitution • Continuing

11. 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. SubOut Integrating Factor, dx • Let the single value, u, represent an algebraic expression in x, say: • Then take thederivative of bothsides • Then Isolate dx

13. SubOut Integrating Factor, dx • Then the Isolated dx: • Thus the SubStitution Components • Consider the previous example • Let: • Then after subbing:

14. 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. SubOut Integrating Factor, dx • Integrating • Recall: • BackSubu=3x+1 into integration result • Expanding the BiNomial find

16. SubOut Integrating Factor, dx • Then • The Same Result as Expanding First then Integrating Term-by-Term Using the Sum Rule

17. GamePlan: 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:

18. GamePlan: 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) • BackSubu = u(x) into H(u) to eliminate u in favor of x to obtain the x-Result:

19. 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. Example: Substitution with e • Try Letting: • Take d/dx of Both Sides • Solving for dx: • Now from u-Definition: • Thendx →

21. 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. 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. Example: Sub Rational Expression • Find • SOLUTION: • Try: • Taking du/dxfind • This produces

24. Example: Sub Rational Expression • Solving • Thus the Answer • An Alternative u:

25. Example: Sub Rational Expression • SubOutxusing: • Find • Then • The Same Result as before

26. 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. 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. Example  DE Model for Annuities • This DE is Variable Separable • Affecting the Separation and Integrating • Find the AntiDerivative by Substitution • Let: • Then:

29. Example  DE Model for Annuities • SubOutA in favor of u: • Integrating:

30. 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 sidesFind the General(Includes C) solution:

31. Example  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 C • Or • Thus the particular soln

32. Example  DE Model for Annuities • Using the Log property • Find • Factoring Out the 40 • Then at 10 years the 403b Amount

33. WhiteBoard Work • Problems From §5.2 • P61 → Retirement Income vs. Outcome • P66 → Price Sensitivity to Supply & Demand

34. All Done for Today SubstitutionCity

35. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

36. ConCavity Sign Chart ConCavityForm ++++++ −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points a b c Inflection NOInflection Inflection