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1 Chapter 8 Techniques of Integration

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2 8.1 Basic Integration Formulas

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4 Example 1 Making a simplifying substitution

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5 Example 2 Completing the square

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6 Example 3 Expanding a power and using a trigonometric identity

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7 Example 4 Eliminating a square root

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8 Example 5 Reducing an improper fraction

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9 Example 6 Separating a fraction

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10 Example 7 Integral of y = sec x

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13 8.2 Integration by Parts

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14 Product rule in integral form Integration by parts formula

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15 Alternative form of Eq. (1) We write

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16 Alternative form of the integration by parts formula

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17 Example 4 Repeated use of integration by parts

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18 Example 5 Solving for the unknown integral

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19 Evaluating by parts for definite integrals or, equivalently

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

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21 Solution

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22 Example 9 Using a reduction formula Evaluate Use

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23 8.3 Integration of Rational Functions by Partial Fractions

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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.

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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.

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26 Example

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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.

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

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29 Integration of rational functions by partial fractions

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30 Example 1 Distinct linear factors

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31 Example 2 A repeated linear factor

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32 Example 3 Integrating an improper fraction

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

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34 Example 5 A repeated irreducible quadratic factor

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

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

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37 8.4 Trigonometric Integrals

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39 Example 1 m is odd

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40 Example 2 m is even and n is odd

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41 Example 3 m and n are both even

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

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43 Example 7 Products of sines and cosines

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44 8.5 Trigonometric Substitutions

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45 Three basic substitutions Useful for integrals involving

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46 Example 1 Using the substitution x=atan

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47 Example 2 Using the substitution x = asin

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48 Example 3 Using the substitution x = asec

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49 Example 4 Finding the volume of a solid of revolution

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50 Solution

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51 8.6 Integral Tables

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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.

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58 8.8 Improper Integrals

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60 Infinite limits of integration

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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?

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63 Solution

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

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

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67 Example 3 Integrands with vertical asymptotes

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68 Example 4 A divergent improper integral Investigate the convergence of

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69 Solution

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70 Example 5 Vertical asymptote at an interior point

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71 Example 5 Vertical asymptote at an interior point

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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.

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

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