Download presentation

Presentation is loading. Please wait.

1
1 Chapter 8 Techniques of Integration

2
2 8.1 Basic Integration Formulas

3
3

4
4 Example 1 Making a simplifying substitution

5
5 Example 2 Completing the square

6
6 Example 3 Expanding a power and using a trigonometric identity

7
7 Example 4 Eliminating a square root

8
8 Example 5 Reducing an improper fraction

9
9 Example 6 Separating a fraction

10
10 Example 7 Integral of y = sec x

11
11

12
12

13
13 8.2 Integration by Parts

14
14 Product rule in integral form Integration by parts formula

15
15 Alternative form of Eq. (1) We write

16
16 Alternative form of the integration by parts formula

17
17 Example 4 Repeated use of integration by parts

18
18 Example 5 Solving for the unknown integral

19
19 Evaluating by parts for definite integrals or, equivalently

20
20 Example 6 Finding area Find the area of the region in Figure 8.1

21
21 Solution

22
22 Example 9 Using a reduction formula Evaluate Use

23
23 8.3 Integration of Rational Functions by Partial Fractions

24
24 General description of the method A rational function f(x)/g(x) can be written as a sum of partial fractions. To do so: (a) The degree of f(x) must be less than the degree of g(x). That is, the fraction must be proper. If it isn ’ t, divide f(x) by g(x) and work with the remainder term. We must know the factors of g(x). In theory, any polynomial with real coefficients can be written as a product of real linear factors and real quadratic factors.

25
25 Reducibility of a polynomial A polynomial is said to be reducible if it is the product of two polynomials of lower degree. A polynomial is irreducible if it is not the product of two polynomials of lower degree. THEOREM (Ayers, Schaum ’ s series, pg. 305) Consider a polynomial g(x) of order n ≥ 2 (with leading coefficient 1). Two possibilities: 1. g(x) = (x-r) h 1 (x), where h 1 (x) is a polynomial of degree n-1, or 2. g(x) = (x 2 +px+q) h 2 (x), where h 2 (x) is a polynomial of degree n-2, and (x 2 +px+q) is the irreducible quadratic factor.

26
26 Example

27
27 Quadratic polynomial A quadratic polynomial (polynomial or order n = 2) is either reducible or not reducible. Consider: g(x)= x 2 +px+q. If (p 2 -4q) ≥ 0, g(x) is reducible, i.e. g(x) = (x+r 1 )(x+r 2 ). If (p 2 -4q) < 0, g(x) is irreducible.

28
28 In general, a polynomial of degree n can always be expressed as the product of linear factors and irreducible quadratic factors:

29
29 Integration of rational functions by partial fractions

30
30 Example 1 Distinct linear factors

31
31 Example 2 A repeated linear factor

32
32 Example 3 Integrating an improper fraction

33
33 Example 4 Integrating with an irreducible quadratic factor in the denominator

34
34 Example 5 A repeated irreducible quadratic factor

35
35 Other ways to determine the coefficients Example 8 Using differentiation Find A, B and C in the equation

36
36 Example 9 Assigning numerical values to x Find A, B and C in

37
37 8.4 Trigonometric Integrals

38
38

39
39 Example 1 m is odd

40
40 Example 2 m is even and n is odd

41
41 Example 3 m and n are both even

42
42 Example 6 Integrals of powers of tan x and sec x

43
43 Example 7 Products of sines and cosines

44
44 8.5 Trigonometric Substitutions

45
45 Three basic substitutions Useful for integrals involving

46
46 Example 1 Using the substitution x=atan

47
47 Example 2 Using the substitution x = asin

48
48 Example 3 Using the substitution x = asec

49
49 Example 4 Finding the volume of a solid of revolution

50
50 Solution

51
51 8.6 Integral Tables

52
52 Integral tables is provided at the back of Thomas ’ T-4 A brief tables of integrals Integration can be evaluated using the tables of integral.

53
53

54
54

55
55

56
56

57
57

58
58 8.8 Improper Integrals

59
59

60
60 Infinite limits of integration

61
61

62
62 Example 1 Evaluating an improper integral on [1,∞] Is the area under the curve y=(ln x)/x 2 from 1 to ∞ finite? If so, what is it?

63
63 Solution

64
64 Example 2 Evaluating an integral on [-∞,∞]

65
65 Using the integral table (Eq. 16) Solution y b 1

66
66

67
67 Example 3 Integrands with vertical asymptotes

68
68 Example 4 A divergent improper integral Investigate the convergence of

69
69 Solution

70
70 Example 5 Vertical asymptote at an interior point

71
71 Example 5 Vertical asymptote at an interior point

72
72 Example 7 Finding the volume of an infinite solid The cross section of the solid in Figure 8.24 perpendicular to the x-axis are circular disks with diameters reaching from the x- axis to the curve y = e x, -∞ < x < ln 2. Find the volume of the horn.

73
73 Example 7 Finding the volume of an infinite solid dx y

Similar presentations

OK

8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals

8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on conceptual art emphasizes Ppt on mobile phones operating systems Ppt on council of ministers of cambodia Ppt on albert einstein biography Ppt on steve jobs biography pdf Org chart download ppt on pollution Edit ppt on google drive Ppt on c++ programming language Ppt on ms excel tutorial Ppt on landing gear systems