jagomart
digital resources
picture1_Study Pdf 92733 | Publication 4 27226 1620


 175x       Filetype PDF       File size 1.05 MB       Source: www.uobabylon.edu.iq


File: Study Pdf 92733 | Publication 4 27226 1620
4100 awl thomas ch08p553 641 8 20 04 10 07 am page 553 chapter 8 techniquesof integration overview the fundamental theorem connects antiderivatives and the definite integral evaluating the indefinite ...

icon picture PDF Filetype PDF | Posted on 17 Sep 2022 | 3 years ago
Partial capture of text on file.
       4100 AWL/Thomas_ch08p553-641  8/20/04  10:07 AM  Page 553
                                Chapter
                                 8  TECHNIQUESOF
                                          INTEGRATION
                                          OVERVIEW The Fundamental Theorem connects antiderivatives and the definite integral.
                                          Evaluating the indefinite integral
                                                                    L ƒsxd dx
                                          is equivalent to finding a function F such that F¿sxd = ƒsxd, and then adding an
                                          arbitrary constant C:
                                                               L ƒsxd dx = Fsxd + C.
                                             In this chapter we study a number of important techniques for finding indefinite
                                          integrals of more complicated functions than those seen before. The goal of this chapter
                                          is to show how to change unfamiliar integrals into integrals we can recognize, find in a
                                          table, or evaluate with a computer. We also extend the idea of the definite integral to
                                          improper integrals for which the integrand may be unbounded over the interval of inte-
                                          gration, or the interval itself may no longer be finite.
                  8.1        Basic Integration Formulas
                                          To help us in the search for finding indefinite integrals, it is useful to build up a table of
                                          integral formulas by inverting formulas for derivatives, as we have done in previous chap-
                                          ters. Then we try to match any integral that confronts us against one of the standard types.
                                          This usually involves a certain amount of algebraic manipulation as well as use of the Sub-
                                          stitution Rule.
                                             Recall the Substitution Rule from Section 5.5:
                                                             L ƒsgsxddg¿sxd dx = L ƒsud du
                                          where u = gsxd is a differentiable function whose range is an interval I and ƒ is continuous
                                          on I. Success in integration often hinges on the ability to spot what part of the integrand
                                          should be called u in order that one will also have du, so that a known formula can be
                                          applied. This means that the first requirement for skill in integration is a thorough mastery of
                                          the formulas for differentiation.
                                                                                                  553
                             Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
             4100 AWL/Thomas_ch08p553-641  8/20/04  10:07 AM  Page 554
                      554           Chapter 8: Techniques of Integration
                                                                                   Table 8.1 shows the basic forms of integrals we have evaluated so far. In this section
                                                                              we present several algebraic or substitution methods to help us use this table. There is a
                                                                              more extensive table at the back of the book; we discuss its use in Section 8.6.
                               TABLE 8.1       Basic integration formulas
                                 1.       du = u + C                                                  13.    cot u du = ln ƒsin uƒ + C
                                      L                                                                     L
                                 2. Lk du = ku + C sany number kd                                                            =-ln ƒcsc uƒ + C
                                                                                                      14. Leu du = eu + C
                                 3.       sdu + dyd =            du +         dy
                                      L                      L           L                            15.       au du = au + C sa 7 0, a Z 1d
                                 4.       un du = un+1 + C sn Z-1d                                          L               ln a
                                      L               n + 1
                                          du                                                          16. L sinh u du = cosh u + C
                                 5. L  u = ln ƒuƒ + C
                                                                                                      17. L cosh u du = sinh u + C
                                 6. L sin u du =-cos u + C
                                                                                                      18.              du        = sin-1 aub + C
                                 7. L cos u du = sin u + C                                                  L2a2 - u2                         a
                                                                                                      19.            du      = 1 tan-1 aub + C
                                 8. L sec2 u du = tan u + C                                                 La2 + u2             a           a
                                                                                                      20.              du          = 1 sec-1 `  u  ` + C
                                 9. L csc2 u du =-cot u + C                                                 Lu2u2 - a2                a            a
                                                                                                      21.              du        = sinh-1 aub + C sa 7 0d
                               10. L sec u tan u du = sec u + C                                             L2a2 + u2                           a
                                                                                                      22.              du        = cosh-1 aub + C su 7 a 7 0d
                               11. L csc u cot u du =-csc u + C                                             L2u2 - a2                           a
                               12.    tan u du =-ln ƒcos uƒ + C
                                      L
                                                      = ln  ƒ sec u ƒ  + C
                                                                             We often have to rewrite an integral to match it to a standard formula.
                                                                             EXAMPLE 1              Making a Simplifying Substitution
                                                                             Evaluate
                                                                                                                                    2x - 9          dx.
                                                                                                                         L2x2 - 9x + 1
                                                           Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
                                4100 AWL/Thomas_ch08p553-641  8/20/04  10:07 AM  Page 555
                                                                                                                                                                                                                                                                                                                                                                               8.1              Basic Integration Formulas                                                                      555
                                                                                                                                                                                                               Solution
                                                                                                                                                                                                                                                                                                    2x - 9                                 dx =                          du                                                                                            2
                                                                                                                                                                                                                                                                           L2x2 - 9x + 1                                                                       L1u                                                                                     u = x - 9x + 1,
                                                                                                                                                                                                                                                                                                                                                                            -1>2                                                                       du = s2x - 9d dx.
                                                                                                                                                                                                                                                                                                                                                    = Lu                                du
                                                                                                                                                                                                                                                                                                                                                    =                us-1>2d+1                             + C                                         Table 8.1 Formula 4,
                                                                                                                                                                                                                                                                                                                                                               s-1>2d + 1                                                                              with n =-1>2
                                                                                                                                                                                                                                                                                                                                                    = 2u1>2 + C
                                                                                                                                                                                                                                                                                                                                                    = 22x2 - 9x + 1 + C
                                                                                                                                                                                                               EXAMPLE 2                                              Completing the Square
                                                                                                                                                                                                               Evaluate
                                                                                                                                                                                                                                                                                                                                                               dx                      .
                                                                                                                                                                                                                                                                                                                                      L28x - x2
                                                                                                                                                                                                               Solution                                We complete the square to simplify the denominator:
                                                                                                                                                                                                                                                                                             2                           2                                                 2
                                                                                                                                                                                                                                                                       8  x - x =-sx - 8xd =-sx - 8x + 16 - 16d
                                                                                                                                                                                                                                                                                                                         2                                                                                                                         2
                                                                                                                                                                                                                                                                                                  =-sx - 8x + 16d + 16 = 16 - sx - 4d .
                                                                                                                                                                                                               Then
                                                                                                                                                                                                                                                                                                                        dx                       =                                           dx
                                                                                                                                                                                                                                                                                               L                                          2              L                                                                 2
                                                                                                                                                                                                                                                                                                           28x - x                                                   216 - sx - 4d
                                                                                                                                                                                                                                                                                                                                                =                                 du                                                           a = 4, u = sx - 4d,
                                                                                                                                                                                                                                                                                                                                                          L                       2                  2                                         du = dx
                                                                                                                                                                                                                                                                                                                                                                      2a - u
                                                                                                                                                                                                                                                                                                                                                = sin-1  aub + C
                                                                                                                                                                                                                                                                                                                                                                                  a                                                            Table 8.1, Formula 18
                                                                                                                                                                                                                                                                                                                                                = sin-1  ax - 4b + C.
                                                                                                                                                                                                                                                                                                                                                                                         4
                                                                                                                                                                                                               EXAMPLE 3                                              Expanding a Power and Using a Trigonometric Identity
                                                                                                                                                                                                               Evaluate
                                                                                                                                                                                                                                                                                                                                                                                     2
                                                                                                                                                                                                                                                                                                                           L ssec x + tan xd  dx.
                                                                                                                                                                                                               Solution                                We expand the integrand and get
                                                                                                                                                                                                                                                                                                                              2                        2                                                                               2
                                                                                                                                                                                                                                                                             ssec x + tan xd = sec  x + 2 sec x tan x + tan  x.
                                                                                                                                                                                                               The first two terms on the right-hand side of this equation are familiar; we can integrate
                                                                                                                                                                                                                                                                                                           2                                                                                                                                                          2
                                                                                                                                                                                                               them at once. How about tan  x?There is an identity that connects it with sec  x:
                                                                                                                                                                                                                                                                                              2                                             2                                     2                             2
                                                                                                                                                                                                                                                                                    tan  x + 1 = sec  x,  tan  x = sec  x - 1.
                                                                                                                                              Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
                                                                              4100 AWL/Thomas_ch08p553-641  8/20/04  10:07 AM  Page 556
                                                                                                                                    556                                                                                 Chapter 8: Techniques of Integration
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          2                                                                           2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    We replace tan  x by sec  x - 1 and get
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   2                                                                                                                      2                                                                                                                                                                                                       2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      ssec x + tan xd  dx =                                                                                                                                                                                          ssec  x + 2 sec x tan x + sec  x - 1d dx
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                L                                                                                                                                                                                                              L
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   = 2L sec2 x dx + 2L sec x tan x dx - L 1 dx
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   = 2 tan x + 2 sec x - x + C.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    EXAMPLE 4                                                                                                                               Eliminating a Square Root
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Evaluate                                                                                                                                                                                                                                                                                         p>421 + cos 4x dx.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               L
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Solution                                                                                             We use the identity
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    cos2 u = 1 + cos 2u,  or 1 + cos 2u = 2 cos2 u.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    With u = 2x, this identity becomes
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          1 + cos 4x = 2 cos2 2x.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Hence,                                                                                                                            p>4                                                                                                                                                                                                               p>4
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              21 + cos 4x dx =                                                                                                                                                                                                  22 2cos2 2x dx
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           0                                                                                                                                                                                                                 0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                L                                                                                                                                                                                                                 L p>4
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      = 22                                                                                                         ƒ    cos 2xƒ  dx                                                                                                                                2u = ƒuƒ
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Lp>4                                                                                                                                                                                          On [0, p>4], cos 2x Ú 0,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      = 22                                                                                                      cos 2x dx                                                                                                                                        so  ƒ cos 2x ƒ                                                                  = cos 2x.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   L                                                                        p>4
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      = 22 csin 2xd                                                                                                                                                                                                                                             Table 8.1, Formula 7, with
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    2                                                                                                                                                           u = 2x and du = 2 dx
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            0                                   22
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      = 22 c1 - 0d =                                                                                                                                                                                              .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  2                                                                                                      2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    EXAMPLE 5                                                                                                                               Reducing an Improper Fraction
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Evaluate
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    3x2 - 7x
                                                                                                                                                                                                                    x - 3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                L  3x + 2  dx.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Solution                                                                                             The integrand is an improper fraction (degree of numerator greater than or
                                                                                                                                                 3x + 23x2 - 7x                                                                                                                                                                                                                                                                                                                    equal to degree of denominator). To integrate it, we divide first, getting a quotient plus a
                                                                                                                                                                                                              3x2 + 2x                                                                                                                                                                                                                                                              remainder that is a proper fraction:
                                                                                                                                                                                                                                                     -9x                                                                                                                                                                                                                                                                                                                                                                                                                                                                      3x2 - 7x                                                                                                                                                                                                           6
                                                                                                                                                                                                                                                     -9x - 6                                                                                                                                                                                                                                                                                                                                                                                                                                                                           3x + 2 = x - 3 + 3x + 2.
                                                                                                                                                                                                                                                                                          + 6                                                                                                                                                                                       Therefore,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      3x2 - 7x dx =                                                                                                                                                     ax - 3 +                                                                                                                6                                     b dx = x2 - 3x + 2 ln  3x + 2 + C.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                L 3x + 2                                                                                                                                                         L                                                                                                                      3x + 2                                                                                                                             2                                                                                                                                  ƒ                                                                   ƒ
                                                                                                                                                                                                                                                                                                                                                                  Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
The words contained in this file might help you see if this file matches what you are looking for:

...Awl thomas chp am page chapter techniquesof integration overview the fundamental theorem connects antiderivatives and definite integral evaluating indefinite l sxd dx is equivalent to finding a function f such that then adding an arbitrary constant c fsxd in this we study number of important techniques for integrals more complicated functions than those seen before goal show how change unfamiliar into can recognize find table or evaluate with computer also extend idea improper which integrand may be unbounded over interval inte gration itself no longer finite basic formulas help us search it useful build up by inverting derivatives as have done previous chap ters try match any confronts against one standard types usually involves certain amount algebraic manipulation well use sub stitution rule recall substitution from section sgsxddg sud du where u gsxd differentiable whose range i continuous on success often hinges ability spot what part should called order will so known formula appl...

no reviews yet
Please Login to review.