Techniques of Integration

9 Key excerpts on "Techniques of Integration"

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  eBook - PDF

  Calculus

  Resequenced for Students in STEM

  • David Dwyer, Mark Gruenwald(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  6 Techniques of Integration 6.1 Advanced Substitution Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 6.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 6.3 Trigonometric Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 6.4 Integrating Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 6.6 Approximating Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Chapter 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Chapter 6 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 6.1 Advanced Substitution Techniques In Section 5.5, we introduced integration by substitution, a technique based on the chain rule for differentiation. In this section we explore this technique in greater depth. Integration by Substitution Revisited In order to become skilled at integration by substitution, it is critical that we be familiar with the basic rules of integration. Table 6.1 lists the rules that we have developed so far.

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  eBook - ePub

  Mathematics for Engineers and Scientists

  • Alan Jeffrey(Author)
  • 2004(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  8

  Systematic integration

  To make effective use of integration it is necessary to develop a systematic approach to the integration of standard forms of integral. This is accomplished in this chapter, which discusses integration by substitution, integration by parts, the use of reduction formulae and the use of partial fractions to simplify the task of integrating rational functions. It is shown how these same techniques may be used to find antiderivatives (indefinite integrals) which are just functions, and definite integrals which are numbers. The chapter ends with a discussion of differentiation under the integral sign and the integration of trigonometric functions involving multiple angles.

  8.1  Integration of elementary functions

  The main objective of this chapter is to explore some of the systematic methods for determining an antiderivative , that is, a function F(x) whose derivative is equal to some given function f(x). As described in the previous chapter, we shall denote the antiderivative of the function f by

  ∫ f

  ( x )

  d x

  with the understanding that

  ∫

  f

  ( x )

  d x = F

  ( x )

  + C

  (8.1)

  with C an arbitrary constant.

  Alternatively, as any indefinite integral of f must also be an anti-derivative of f , we may identify F(x) in Eqn (8.1) with

  ∫ a x

  f

  ( t )

  dy , where a is arbitrary, and incorporate the constant C into the constant resulting from the arbitrary lower limit to obtain the equivalent expression

  ∫

  f

  ( x )

  dx =

  ∫ a x

  f

  ( t )

  d t .

  (8.2)

  Remember that the symbol

  ∫ f

  ( x )

  d x

  is derived from differentiation and denotes the most general function whose derivative is f . The allied symbol

  ∫

  a ˙

  b

  f

  ( x )

  d x

  denoting a definite integral of f , derives from integration and is simply a real number.

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  eBook - PDF

  Mathematics N5 Student's Book

  TVET FIRST

  • JV John(Author)
  • 2022(Publication Date)
  • Macmillan

    (Publisher)

  In this module, we will learn some of the elementary methods of integration, including inspection, algebraic and trigonometric substitutions, using partial fractions and integration by parts. Note The full learning outcomes for each module are listed in the table at the back of the book. Integration techniques Types: o ∫[ f (x) ]nf '(x) dx = [ f (x) ] n + 1 _ n + 1 + c o ∫ f '(x) _ f (x) dx = ln f (x) + c o ∫ a f (x) f '(x) dx o ∫ e f (x) f '(x) dx Other substitutions 4.3: Integration by algebraic substitution Introduction to the indefinite integral Methods of integration Standard integrals Trigonometric identities 4.1: Revision of basic integration Readily determining the integral directly by examining the integrand 4.2: Integration by inspection Squares of sin, cos, tan and cot Products of powers of sin and cos: ∫ sin m x cos n x dx, m or n odd or both odd Powers of tan or cot Products of sin and/or cos with different coefficients 4.4: Integration of trigonometric functions Finding integrals using trigonometric substitution Using trigonometric substitution to derive the formulae: o ∫ 1 ___________ √ _ a 2 − b 2 x 2 dx = 1 _ b sin −1 bx _ a + c o ∫ √ _ a 2 − b 2 x 2 dx = a 2 _ 2b sin −1 bx _ a + x _ 2 √ _ a 2 − b 2 x 2 + c o ∫ 1 _ a 2 + b 2 x 2 dx = 1 _ ab tan −1 bx _ a + c Using the formulae with or without the b factor 4.5: Trigonometric substitution Suitable substitution Substitution after long division 4.6: Integration of algebraic fractions Integrating the following types of fractions using partial fractions: o ∫ f (x) ___________ (ax ± b)(cx ± d) dx o ∫ f (x) _ (ax ± b) n dx, n = 2 or 3 4.7: Integration using partial fractions Integrating products of two functions that are not the derivatives of each other using the formula: ∫ f (x) g'(x) dx = f (x) g(x) − ∫ f '(x) g(x) dx 4.8: Integration by parts Figure 4.1: Integration is an essential tool for engineers

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  Differentiation and Integration

  • W. Bolton(Author)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  6.1 Basic rules 6 Techniques of Integration This chapter is about basic rules and techniques that are involved in integration. This opening section is about the basic rules with the following sections demonstrating techniques commonly used to rearrange functions so that they can be integrated. These are: substitutions, integration by parts and partial fractions. These rules and techniques are used in applications involving chapters 7, 8 and 9. This chapter can thus be regarded as introducing the rules and techniques illustrated with examples and problems for practice, these being: 1 Function multiplied by a constant ( 6.1.1. ). 2 Sum of functions (6.1.2}. 3 The technique of substitution (6.2). 4 The technique of integration by parts (6.3}. 5 The technique of integration by partial fractions (6.4). 6.1.1 Multiplication by a constant The derivative, for example, of 5r is 5 multiplied by the derivative of r, i.e. lOx. The derivative of aj(x), where a is a constant, is (see section 2.1.2} Thus the integral of aj(x) is the constant a multiplied by the integral ofj(x), i.e. J aj(x) dx =a jf(x) dx [1] 107 108 DlFFERENTIATION AND INTEGRATION Example Evaluate the following integrals: (a) J 4x 2 dx, (b) J 2cosxdx, (c) J ~dx (a) Using the above rule for the multiplication by a constant, J 4x 2 dx = 4 J x 2 dx = tx 3 + C (b) Using the rule for the multiplication by a constant, J 2cosxdx= 2 J cosxdx = 2sinx+C (c) Using the rule for the multiplication by a constant. J~dx=4 J ~dx=4lnx+C Sometimes it is convenient to write this as 4 In x + In A, where the constant Cis written in terms of another constant A, i.e.

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  What is Calculus?

  From Simple Algebra to Deep Analysis

  • R Michael Range(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  Mathematica ) can handle such computations much more efficiently and quickly, allowing the student to focus on understanding the essential ideas rather than getting lost in a multitude of special cases and techniques. Experience and practice are helpful, but there are limits even for the most experienced mathematician, since a particular problem may not have an explicit answer at all in terms of known functions. So, no matter how hard one tries to apply known techniques and tricks, it may all be of no use... .

  IV.8.6     Exercises

  1.   Find explicit formulas for the antiderivatives

  a)  ∫ x cos x dx ;

  b)  ∫ x 2 cos x dx ;

  c)  ∫ x 3

  ex dx

  ;

  d)  ∫ t 2t dt .

  2.   Find ∫ sin

  t et dt

  .

  3.   Find ∫ x 2 ln x dx . (Hint: Compare with ∫ ln x dx in the text.)

  4.   Find an antiderivative of y = arcsin x as follows.

  a)  Apply integration by parts to ∫ arcsin x · 1 dx , with g ′ = 1. (See Section II.6.5 for the derivative of y = arcsin x .)

  b)  Use the substitution u = 1 − x 2 in the remaining integral.

  IV.9     Higher Order Approximations, Part 2: Taylor’s Theorem

  In Section III.9 we had considered approximations of functions by so-called Taylor polynomials. The discussion culminated with some remarkable new representations for the exponential and trigonometric functions. The reader should briefly review the earlier discussion. At that time we accepted the main results based on intuitive principles and graphical evidence. We are now in a position to use integrals to formulate a precise formula for the error between a function and its Taylor polynomials. This result is known as Taylor’s Theorem with Remainder

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  Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  But it may not be obvious which technique we should use to integrate a given function. ■ Guidelines for Integration Until now individual techniques have been applied in each section. For instance, we usu- ally used substitution in Exercises 4.5, integration by parts in Exercises 7.1, and partial fractions in Exercises 7.4. But in this section we present a collection of miscellaneous integrals in random order and the main challenge is to recognize which technique or formula to use. No hard and fast rules can be given as to which method applies in a given situation, but we give some general guidelines that you may find useful. A prerequisite for applying a strategy is a knowledge of the basic integration formu- las. In the following table we have collected the integrals from our previous list together with several additional formulas that we have learned in this chapter. Table of Integration Formulas Constants of integration have been omitted. 1. y x n dx - x n11 n 1 1 sn ± 21d 2. y 1 x dx - ln | x | 3. y e x dx - e x 4. y b x dx - b x ln b 5. y sin x dx - 2cos x 6. y cos x dx - sin x 7. y sec 2 x dx - tan x 8. y csc 2 x dx - 2cot x 9. y sec x tan x dx - sec x 10. y csc x cot x dx - 2csc x 11. y sec x dx - ln | sec x 1 tan x | 12. y csc x dx - ln | csc x 2 cot x | 13. y tan x dx - ln | sec x | 14. y cot x dx - ln | sin x | 15. y sinh x dx - cosh x 16. y cosh x dx - sinh x 17. y dx x 2 1 a 2 - 1 a tan 21 S x a D 18. y dx sa 2 2 x 2 - sin 21 S x a D , a . 0 *19. y dx x 2 2 a 2 - 1 2a ln Z x 2 a x 1 a Z *20. y dx sx 2 6 a 2 - ln | x 1 sx 2 6 a 2 | 7.5 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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  Calculus: Early Transcendentals, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  But it may not be obvious which technique we should use to integrate a given function. ■ Guidelines for Integration Until now individual techniques have been applied in each section. For instance, we usu- ally used substitution in Exercises 5.5, integration by parts in Exercises 7.1, and partial fractions in Exercises 7.4. But in this section we present a collection of miscellaneous integrals in random order and the main challenge is to recognize which technique or formula to use. No hard and fast rules can be given as to which method applies in a given situation, but we give some general guidelines that you may find useful. A prerequisite for applying a strategy is a knowledge of the basic integration formu- las. In the following table we have collected the integrals from our previous list together with several additional formulas that we have learned in this chapter. Table of Integration Formulas Constants of integration have been omitted. 1. y x n dx - x n11 n 1 1 sn ± 21d 2. y 1 x dx - ln | x | 3. y e x dx - e x 4. y b x dx - b x ln b 5. y sin x dx - 2cos x 6. y cos x dx - sin x 7. y sec 2 x dx - tan x 8. y csc 2 x dx - 2cot x 9. y sec x tan x dx - sec x 10. y csc x cot x dx - 2csc x 11. y sec x dx - ln | sec x 1 tan x | 12. y csc x dx - ln | csc x 2 cot x | 13. y tan x dx - ln | sec x | 14. y cot x dx - ln | sin x | 15. y sinh x dx - cosh x 16. y cosh x dx - sinh x 17. y dx x 2 1 a 2 - 1 a tan 21 S x a D 18. y dx sa 2 2 x 2 - sin 21 S x a D , a . 0 *19. y dx x 2 2 a 2 - 1 2a ln Z x 2 a x 1 a Z *20. y dx sx 2 6 a 2 - ln | x 1 sx 2 6 a 2 | 7.5 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  C h a p te r 18 Techniques of Integration, Part One Let’s kick off the process of building up a virtual toolkit of techniques to find antiderivatives. In this chapter, we’ll look at the following three techniques: • the method of substitution (otherwise known as “change of variables”); • integration by parts; and • using partial fractions to integrate rational functions. Then, in the next chapter, we’ll look at some more techniques involving trig functions. 18.1 Substitution Using the chain rule, we can easily differentiate e x 2 with respect to x and see that d dx e x 2 = 2 xe x 2 . The factor 2 x is the derivative of x 2 , which appears in the exponent. Now, as we saw in Section 17.4 of the previous chapter, we can flip the equation around to get Z 2 xe x 2 dx = e x 2 + C for some constant C . So we can integrate 2 xe x 2 with respect to x . How about just e x 2 ? You’d think it would be just as easy, if not easier, to find Z e x 2 dx. It turns out that it’s not just hard to find this—it’s impossible! Well, not quite impossible, but the fact is, there’s no “nice” expression for an antiderivative of e x 2 . (You have to resort to infinite series, definite integrals, or some other sort of roundabout device.) Perhaps you think that e x 2 / 2 x works? Nope—use the quotient rule to differentiate this (with respect to x ) and you’ll see that you get something quite different from e x 2 . 384 • Techniques of Integration, Part One What saves us in the case of R 2 xe x 2 dx is the presence of the 2 x factor, which is exactly what popped out when we used the chain rule to differentiate e x 2 . Now, imagine starting with an indefinite integral like this: Z x 2 cos( x 3 ) dx. We’re taking the cosine of the somewhat nasty quantity x 3 , but there’s a ray of hope: the derivative of this quantity is 3 x 2 . This almost matches the factor x 2 in the integrand—it’s only the constant 3 that makes things a little more difficult.

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  Mathematics N5 Student's Book

  TVET FIRST

  • GI Mapaling(Author)
  • 2016(Publication Date)
  • Troupant

    (Publisher)

  94 M O D U L E Integration techniques 4 Think about it 1. What basic integration formulas do you know? 2. What procedures do you know for matching integrals to basic formulas? 3. What is tabular integration? Unit 4.1: Integration by inspection When dealing with integration, we use a standard list of results from differentiation to guide us in finding certain integrals. Overview On completion of this unit, you should be able to identify and determine the integrals of the following forms: • ∫ [ f ( x ) ] n . f ′ ( x ) dx • ∫ f ′ ( x ) ____ f ( x ) dx • ∫ e f(x ) . f ′ ( x ) dx • ∫ a f ( x ) . f ′ ( x ) dx Pre-knowledge General rules f ( x ) ∫ f ( x ) dx Mathematical tool k kx + c Constant multiple rule kx n [ n ≠ − 1] kx n + 1 _____ n + 1 + c Power rule kx − 1 = k __ x k ln x + c Power rule e kx e kx ___ k + c Chain rule a kx a kx _____ k ln a + c Chain rule sin kx − cos kx _____ k + c Chain rule cos kx sin kx _____ k + c Chain rule 95 Pre-knowledge (continued) f ( x ) ∫ f ( x ) dx Mathematical tool tan kx ln sec kx _______ k + c Chain rule cot kx ln sin kx _______ k + c Chain rule sec kx tan kx sec kx _____ k + c Chain rule cosec kx cot kx − cosec kx _______ k + c Chain rule sec 2 kx tan kx _____ k + c Chain rule cosec 2 kx − cot kx _____ k + c Chain rule 1 _______ √ _____ 1 − x 2 arcsin x + c Inverse trigonometric functions − 1 _______ √ _____ 1 − x 2 arccos x + c Inverse trigonometric functions 1 _____ 1 + x 2 arctan x + c Inverse trigonometric functions − 1 _____ 1 + x 2 arccot x + c Inverse trigonometric functions 1 ________ x √ _____ x 2 − 1 arcsec x + c Inverse trigonometric functions − 1 ________ x √ _____ x 2 − 1 arccosec x + c Inverse trigonometric functions Type 1 : ∫ [ f ( x ) ] n . f ′ ( x ) dx = [ f ( x ) ] n + 1 _______ _ n + 1 + c In order to apply this formula, we need the following components in the integral: 1. f ( x ) raised to some positive, negative or surd (fractional) numerical index.

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