ESSENTIAL CALCULUS CH06 Techniques of integration

1. ESSENTIAL CALCULUSCH06 Techniques of integration

2. In this Chapter: • 6.1 Integration by Parts • 6.2 Trigonometric Integrals and Substitutions • 6.3 Partial Fractions • 6.4 Integration with Tables and Computer Algebra Systems • 6.5 Approximate Integration • 6.6 Improper Integrals Review

3. Chapter 6, 6.1, P307

4. Chapter 6, 6.1, P307

5. Chapter 6, 6.1, P309

6. ▓How to Integrate Powers of sin x and cos x From Examples 1– 4 we see that the following strategy works: Chapter 6, 6.2, P314

7. (i) If the power of cos x is odd, save one cosine factor and use cos2x=1-sin2x to express the remaining factors in terms of sin x. Then substitute u=sin x. Chapter 6, 6.2, P314

8. (ii) If the power of sin x is odd, save one sine factor and use sin2x=1-cos2x to express the remaining factors in terms of cos x. Then substitute u=cos x. Chapter 6, 6.2, P314

9. (iii) If the powers of both sine and cosine are even, use the half-angle identities: It is sometimes helpful to use the identity Chapter 6, 6.2, P314

10. ▓How to Integrate Powers of tan x and sec x From Examples 5 and 6 we have a strategy for two cases Chapter 6, 6.2, P315

11. (i) If the power of sec x is even, save a factor of sec2x and use sec2x=1+tan2x to express the remaining factors in terms of tan x. Then substitute u=tan x. Chapter 6, 6.2, P315

12. (ii) If the power of tan x is odd, save a factor of sec x tan x and use tan2x=sec2x-1 to express the remaining factors in terms of sec x. Then substitute u sec x. Chapter 6, 6.2, P315

13. Chapter 6, 6.2, P315

14. Chapter 6, 6.2, P315

15. TABLE OF TRIGONOMETRIC SUBSTITUTIONS Expression Substitution Identity Chapter 6, 6.2, P317

16. Chapter 6, 6.3, P326

17. Chapter 6, 6.5, P336

18. Chapter 6, 6.5, P336

19. Chapter 6, 6.5, P336

20. If we divide [a,b] into n subintervals of equal length ∆x=(b-a)/n , then we have where X*1 is any point in the ith subinterval [xi-1,xi]. Chapter 6, 6.5, P336

21. Left endpoint approximation Chapter 6, 6.5, P336

22. Right endpoint approximation Chapter 6, 6.5, P336

23. MIDPOINT RULE where and Chapter 6, 6.5, P336

24. Chapter 6, 6.5, P337

25. TRAPEZOIDAL RULE Where ∆x=(b-a)/n and xi=a+i∆x. Chapter 6, 6.5, P337

26. Chapter 6, 6.5, P337

27. 3. ERROR BOUNDS Suppose │f”(x)│≤K for a≤x≤b. If ET and EM are the errors in the Trapezoidal and Midpoint Rules, then and Chapter 6, 6.5, P339

28. Chapter 6, 6.5, P340

29. Chapter 6, 6.5, P340

30. SIMPSON’S RULE Where n is even and ∆x=(b-a)/n. Chapter 6, 6.5, P342

31. ERROR BOUND FOR SIMPSON’S RULE Suppose that │f(4)(x)│≤K for a≤x≤b. If Es is the error involved in using Simpson’s Rule, then Chapter 6, 6.5, P343

32. In defining a definite integral we dealt with a function f defined on a finite interval [a,b]. In this section we extend the concept of a definite integral to the case where the interval is infinite and also to the case where f has an infinite discontinuity in [a,b]. In either case the integral is called an improper integral. Chapter 6, 6.6, P347

33. Improper integrals: Type1: infinite intervals Type2: discontinuous integrands Chapter 6, 6.6, P347

34. DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1 (a) If exists for every number t≥a, then provided this limit exists (as a finite number). (b) If exists for every number t≤b, then provided this limit exists (as a finite number). Chapter 6, 6.6, P348

35. The improper integrals and are called convergent if the corresponding limit exists and divergent if the limit does not exist. (c) If both and are convergent, then we define In part (c) any real number can be used (see Exercise 52). Chapter 6, 6.6, P348

36. is convergent if p>1 and divergent if p≤1. Chapter 6, 6.6, P351

37. 3.DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2 • If f is continuous on [a,b) and is discontinuous at b, then • if this limit exists (as a finite number). • (b) If f is continuous on (a,b] and is discontinuous at a, then • if this limit exists (as a finite number). Chapter 6, 6.6, P351

38. The improper integral is called convergent if the corresponding limit exists and divergent if the limit does not exist (c)If f has a discontinuity at c, where a<c<b, and both and are convergent, then we define Chapter 6, 6.6, P351

39. erroneous calculation: This is wrong because the integral is improper and must be calculated in terms of limits. Chapter 6, 6.6, P352

40. COMPARISON THEOREM Suppose that f and g are continuous functions with f(x)≥g(x)≥0 for x≥a . • (b) If is divergent, then is divergent. • If is convergent, then is convergent. Chapter 6, 6.6, P353