Calculus- Early Transcendentals, 2021a

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278 Techniques of Integration Find the antiderivatives. ∫ Exercise 7.4.15 xsinx dx Exercise 7.4.29 ∫ xtan −1 xdx Exercise 7.4.16 ∫ xe −x dx Exercise 7.4.30 ∫ sin −1 xdx Exercise 7.4.17 ∫ x 2 sinxdx Exercise 7.4.31 ∫ xlnxdx Exercise 7.4.18 ∫ x 3 sinxdx Exercise 7.4.32 ∫ (x − 2)lnxdx Exercise 7.4.19 ∫ xe x2 dx Exercise 7.4.33 ∫ xln(x − 1) dx Exercise 7.4.20 ∫ x 3 e x dx Exercise 7.4.34 ∫ xln(x 2 ) dx Exercise 7.4.21 ∫ xe −2x dx Exercise 7.4.35 ∫ x 2 lnx dx Exercise 7.4.22 ∫ e x sinx dx Exercise 7.4.36 ∫ (lnx) 2 dx Exercise 7.4.23 ∫ e 2x cosx dx Exercise 7.4.37 ∫ (ln(x + 1)) 2 dx Exercise 7.4.24 ∫ e 2x sin(3x) dx Exercise 7.4.38 ∫ xsec 2 xdx Exercise 7.4.25 ∫ e 5x cos(5x) dx Exercise 7.4.39 ∫ xcsc 2 xdx Exercise 7.4.26 ∫ sinxcosxdx Exercise 7.4.40 ∫ x √ x − 2 dx Exercise 7.4.27 ∫ sin −1 xdx Exercise 7.4.41 ∫ xsecxtanxdx Exercise 7.4.28 ∫ tan −1 (2x) dx Exercise 7.4.42 ∫ xcscxcotxdx Evaluate the indefinite integral after first making a substitution. ∫ Exercise 7.4.43 sin(lnx) dx Exercise 7.4.45 ∫ ln( √ x) dx Exercise 7.4.44 ∫ sin( √ x) dx Exercise 7.4.46 ∫ e √x dx
7.5. Rational Functions 279 Exercise 7.4.47 ∫ e lnx dx Evaluate the definite integral. Exercise 7.4.48 ∫ π 0 xsinx dx Exercise 7.4.53 ∫ 1 0 x 3 e x dx Exercise 7.4.49 Exercise 7.4.50 Exercise 7.4.51 ∫ 1 −1 ∫ π/4 −π/4 ∫ π/2 −π/2 xe −x dx x 2 sinxdx x 3 sinxdx Exercise 7.4.54 Exercise 7.4.55 ∫ 2 1 ∫ π 0 xe −2x dx e x sinx dx Exercise 7.4.52 ∫ √ ln2 0 xe x2 dx Exercise 7.4.56 ∫ π/2 −π/2 e 2x cosx dx Exercises 7.4.15 to 7.4.56 were adapted by Lyryx from APEX <strong>Calculus</strong>, Version 3.0, written by G. Hartman. This material is released under Creative Commons license CC BY-NC (https://creativecommons.org/licenses/by-nc/4.0/ ). See the Copyright and Revision History pages in the front of this text for more information. 7.5 Rational Functions A rational function is a fraction with polynomials in the numerator and denominator. For example, x 3 x 2 + x − 6 , 1 (x − 3) 2 , x 2 + 1 x 2 − 1 , are all rational functions of x. There is a general technique called “partial fractions” that, in principle, allows us to integrate any rational function. The algebraic steps in the technique are rather cumbersome if the polynomial in the denominator has degree more than 2, and the technique requires that we factor the denominator, something that is not always possible. However, in practice one does not often run across rational functions with high degree polynomials in the denominator for which one has to find the antiderivative function. So we shall explain how to find the antiderivative of a rational function only when the denominator is a quadratic polynomial ax 2 + bx + c. We should mention a special type of rational function that we already know how to integrate: If the denominator has the form (ax + b) n , the substitution u = ax + b will always work. The denominator becomes u n , and each x in the numerator is replaced by (u−b)/a,anddx = du/a. While it may be tedious to complete the integration if the numerator has high degree, it is merely a matter of algebra. Example 7.32: Substitution and Splitting Up a Fraction ∫ Find x 3 (3 − 2x) 5 dx.
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with Open Texts Calculus Early Tran
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Calculus: Early Transcendentals an
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Calculus: Early Transcendentals an
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Table of Contents Table of Contents
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v 5.4.3 Taylor Polynomials . . . .
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vii 13.2 Limits and Continuity . .
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Introduction The emphasis in this c
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4 Review In the table, the set of r
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6 Review 1.1.3 The Quadratic Formul
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8 Review 1.1.4 Inequalities, Interv
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10 Review Inequality Rules Add/subt
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12 Review Guidelines for Solving Ra
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14 Review Looking where the
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16 Review Example 1.17: Absolute Va
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18 Review (a) |x|≥2 (b) |x − 3|
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20 Review The most familiar form of
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22 Review Example 1.25: Equa
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24 Review |Δy|, as shown in figure
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26 Review • (h,k) is the vert
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28 Review Determining the Type of C
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30 Review To determine a, we substi
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32 Review (a) A(2,0),B(4,3) (b) A(
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34 Review • Secant (abbreviated b
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36 Review Reading from the unit cir
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38 Review Notice that we can now fi
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40 Review 1.3.4 Graphs of Trigonome
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42 Review Exercises for 1.3 Exercis
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44 Review Exercise 1.4.7 Simplify t
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46 Functions Example 2.2: Domain of
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48 Functions Example 2.5: Domain Fi
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50 Functions For horizontal and ver
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52 Functions Solution. The domain o
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54 Functions After the positive int
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56 Functions Example 2.11: Determin
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58 Functions Example 2.16: Finding
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60 Functions Since the function f (
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62 Functions Example 2.21: Solve Lo
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64 Functions We can do something si
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66 Functions Cancellation Rules sin
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68 Functions (a) sin −1 ( √ 3/2
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70 Functions Example 2.32: Computin
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72 Functions (a) f (x) (b) − f (x
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Chapter 3 Limits 3.1 The Limit The
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3.2. Precise Definition of a Limit
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3.2. Precise Definition of a Limit
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3.3. Computing Limits: Graphically
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3.4. Computing Limits: Algebraicall
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3.4. Computing Limits: Algebraicall
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3.5. Infinite Limits and Limits at
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5 + x −1 (n) lim x→∞ 1 + 2x
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3.6. A Trigonometric Limit 99 Figur
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3.6. A Trigonometric Limit 101 Solu
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3.7. Continuity 103 § § ¥
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3.7. Continuity 105 Definition 3.43
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3.7. Continuity 107 Definition 3.45
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3.7. Continuity 109 Solution. We wi
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3.7. Continuity 111 Therefore, our
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3.7. Continuity 113 Exercise 3.7.3
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116 Derivatives doesn’t meet the
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118 Derivatives of the slope of the
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120 Derivatives algebra to find a s
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122 Derivatives we often use f and
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124 Derivatives you are required to
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126 Derivatives We can summarize
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128 Derivatives 4.2.3 Velocities Su
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130 Derivatives Exercise 4.2.5 Find
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132 Derivatives Proof. For convenie
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134 Derivatives Exercises for Secti
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136 Derivatives since sinx cosx −
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138 Derivatives In practice, of cou
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140 Derivatives Exercises for Secti
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142 Derivatives Exercise 4.5.39 Fin
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144 Derivatives . . . . Figure 4.4:
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146 Derivatives = ( d dx x2 ln2) e
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148 Derivatives Example 4.38: Deriv
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150 Derivatives Example 4.42: Equat
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152 Derivatives Solution. We take l
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154 Derivatives Exercise 4.7.8 Find
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156 Derivatives 4.8.1 Derivatives o
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158 Derivatives Exercise 4.8.4 The
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Chapter 5 Applications of Derivativ
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5.1. Related Rates 163 Solution. He
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5.1. Related Rates 165 Example 5.6:
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5.2. Extrema of a Function 167 Exer
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5.2. Extrema of a Function 169 poin
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5.2. Extrema of a Function 171 Exer
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5.2. Extrema of a Function 173 3. E
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5.2. Extrema of a Function 175 A si
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5.3. The Mean Value Theorem 177 2.
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5.3. The Mean Value Theorem 179
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5.3. The Mean Value Theorem 181 Exe
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5.4. Linear and Higher Order Approx
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5.5. L’Hôpital’s Rule 191 "bou
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5.5. L’Hôpital’s Rule 193 Exam
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5.5. L’Hôpital’s Rule 195 Exer
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5.6. Curve Sketching 197 consistent
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5.6. Curve Sketching 199 Solution.
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5.6. Curve Sketching 201 the concav
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5.6. Curve Sketching 203 If there a
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5.6. Curve Sketching 205 Note that
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5.7. Optimization Problems 207 Guid
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5.7. Optimization Problems 209 is t
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5.7. Optimization Problems 211 (Alt
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5.7. Optimization Problems 213 Exer
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Chapter 6 Integration 6.1 Displacem
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6.1. Displacement and Area 217 Exam
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6.1. Displacement and Area 219 draw
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6.1. Displacement and Area 221 It i
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6.1. Displacement and Area 223
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6.1. Displacement and Area 225 We d
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Page 318 and 319: 302 Applications of Integration For
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328 Applications of Integration Exe
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330 Applications of Integration Unf
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332 Applications of Integration Fig
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334 Applications of Integration is
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Chapter 9 Sequences and Series Cons
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9.1. Sequences 339 log 2 (1 + ε) >
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9.1. Sequences 341 Example 9.8: Con
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9.1. Sequences 343 below it is boun
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9.2. Series 345 If {kx n } ∞ n=0
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9.2. Series 347 for some number s.
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9.3. The Integral Test 349 If all o
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9.3. The Integral Test 351 Proof. W
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9.4. Alternating Series 353 Exercis
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9.5. Comparison Tests 355 Exercises
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9.5. Comparison Tests 357 Solution.
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9.7. The Ratio and Root Tests 359 E
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9.7. The Ratio and Root Tests 361 P
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9.8. Power Series 363 Definition 9.
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Exercise 9.8.2 Find the radius of c
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f ′′ (x)= f ′′′ (x)= ∞
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9.10. Taylor Series 369 Solution. T
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9.11. Taylor’s Theorem 371 a posi
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9.11. Taylor’s Theorem 373 Note t
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Chapter 10 Differential Equations M
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10.1. First Order Differential Equa
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10.1. First Order Differential Equa
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10.2. First Order Homogeneous Linea
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10.3. First Order Linear Equations
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10.4. Approximation 385 Exercises f
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10.4. Approximation 387 y .. 0.5 .
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10.5. Second Order Homogeneous Equa
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10.5. Second Order Homogeneous Equa
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10.6. Second Order Linear Equations
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Chapter 11 Polar Coordinates, Param
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11.1. Polar Coordinates 403 Solutio
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11.2. Slopes in Polar Coordinates 4
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11.3. Areas in Polar Coordinates 40
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11.3. Areas in Polar Coordinates 40
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11.4. Parametric Equations 411 Figu
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11.5 Calculus with Parametric Equat
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11.6. Conics in Polar Coordinates 4
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11.6. Conics in Polar Coordinates 4
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Chapter 12 Three Dimensions 12.1 Th
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12.1. The Coordinate System 421 Now
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12.2. Vectors 423 12.2 Vectors A ve
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12.2. Vectors 425 3.54 0.77 . . . .
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12.3. The Dot Product 427 Exercise
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12.3. The Dot Product 429 Solution.
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12.3. The Dot Product 431 v .. v ..
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12.4. The Cross Product 433 Exercis
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12.4. The Cross Product 435 We know
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12.5. Lines and Planes 437 Exercise
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12.5. Lines and Planes 439 Solution
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12.5. Lines and Planes 441 Example
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12.5. Lines and Planes 443 Exercise
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12.6. Other Coordinate Systems 445
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452 Partial Differentiation the lin
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454 Partial Differentiation Exercis
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456 Partial Differentiation Fortuna
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460 Partial Differentiation paralle
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462 Partial Differentiation 3 z . 2
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466 Partial Differentiation lim ε
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468 Partial Differentiation 13.5 Di
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470 Partial Differentiation Example
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476 Partial Differentiation so we g
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478 Partial Differentiation 0.0 0.2
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480 Partial Differentiation and the
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482 Partial Differentiation xz = 2y
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484 Partial Differentiation x φ .
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486 Multiple Integration point mult
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488 Multiple Integration Figure 14.
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490 Multiple Integration 1 0 0 1 Fi
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492 Multiple Integration Exercise 1
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496 Multiple Integration This examp
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498 Multiple Integration Exercise 1
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500 Multiple Integration Finally, M
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504 Multiple Integration Example 14
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506 Multiple Integration ∫ 1 = 1
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508 Multiple Integration Solution.
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510 Multiple Integration ∫ ∫
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.. 512 Multiple Integration As befo
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514 Multiple Integration y . ......
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516 Multiple Integration ∫∫ Exe
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518 Vector Functions separate funct
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520 Vector Functions = 〈 f ′ (t
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522 Vector Functions Figure 15.6:
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524 Vector Functions 2.0 1.5 1.0 0.
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526 Vector Functions Exercise 15.2.
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528 Vector Functions (This integral
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530 Vector Functions To remove the
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532 Vector Functions Graphing this
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534 Vector Functions Example 15.20
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536 Vector Calculus which is the re
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538 Vector Calculus Since f x = 2e
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540 Vector Calculus Definition 16.4
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542 Vector Calculus Example 16.8: W
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544 Vector Calculus Proof. We write
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546 Vector Calculus Exercise 16.3.2
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548 Vector Calculus In this case, n
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550 Vector Calculus We can now rewr
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552 Vector Calculus 16.5 The Diverg
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554 Vector Calculus The remaining f
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556 Vector Calculus Suppose we inst
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558 Vector Calculus circle of radiu
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560 Vector Calculus Exercise 16.6.7
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562 Vector Calculus we might want t
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564 Vector Calculus where e is an e
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566 Vector Calculus ∫ b = ∫ = a
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Selected Exercise Answers 1.1.1 1.
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571 2.8.4 1. [2,3) ∪ (3,∞) 2. (
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573 4.5.18 −3(4 − x) 2 4.5.19 6
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575 5.1.15 500/ √ 3 − 200 ≈ 8
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577 5.6.21 min at x = 1 5.6.22 none
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579 7.1.2 x 5 /5 + 2x 3 /3 + x +C 7
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581 7.4.19 1/2e x2 +C 7.4.20 x 3 e
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583 7.7.20 diverges 7.7.21 diverges
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585 8.6.5 ¯x = 45/28, ȳ = 93/70 8
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587 9.8.2 R = e 10.1.2 y = arctant
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589 10.6.8 Ae t + Be 3t +(1/2)te 3t
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591 12.5.2 4(x + 1)+5(y − 2) −
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593 13.6.5 f x = 3cos(3x)cos(2y), f
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595 14.3.6 ¯x = 6/5, ȳ = 12/5 14.
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597 15.5.8 〈−3sint,2cost + 1/10
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Index absolute extrema, 172, 206 ab
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INDEX 601 at infinity, 87 indetermi
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Table of Integrals Table of Integra
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∫ 41. sin n ucos m udu= − sinn
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Integrals Involving u 2 − a 2 , a
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∫ du 108. u √ a + bu = √ 1
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