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1 Chapter 8 Techniques of Integration. 2 8.1 Basic Integration Formulas.

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Presentation on theme: "1 Chapter 8 Techniques of Integration. 2 8.1 Basic Integration Formulas."— Presentation transcript:

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


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