1. ESSENTIAL CALCULUS CH06 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

5. Chapter 6, 6.1, P309

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

7. Chapter 6, 6.2, P314 (i) If the power of cos x is odd, save one cosine factor and use cos 2 x=1-sin 2 x to express the remaining factors in terms of sin x. Then substitute u=sin x.

8. Chapter 6, 6.2, P314 (ii) If the power of sin x is odd, save one sine factor and use sin 2 x=1-cos 2 x to express the remaining factors in terms of cos x. Then substitute u=cos x.

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

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

11. Chapter 6, 6.2, P315 (i) If the power of sec x is even, save a factor of sec 2 x and use sec 2 x=1+tan 2 x to express the remaining factors in terms of tan x. Then substitute u=tan x.

12. Chapter 6, 6.2, P315 (ii) If the power of tan x is odd, save a factor of sec x tan x and use tan 2 x=sec 2 x-1 to express the remaining factors in terms of sec x. Then substitute u sec x.

13. Chapter 6, 6.2, P315

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

16. Chapter 6, 6.3, P326

17. 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 [x i-1,x i ].

21. Chapter 6, 6.5, P336 Left endpoint approximation

22. Chapter 6, 6.5, P336 Right endpoint approximation

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

24. Chapter 6, 6.5, P337

25. TRAPEZOIDAL RULE Where ∆x=(b-a)/n and x i =a+i∆x.

26. Chapter 6, 6.5, P337

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

28. Chapter 6, 6.5, P340

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

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

32. Chapter 6, 6.6, P347 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.

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

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).

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

37. Chapter 6, 6.6, P351 3.DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2 (a)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).

38. Chapter 6, 6.6, P351 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

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

40. Chapter 6, 6.6, P353 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. (a) If is convergent, then is convergent.