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Calculus Volume 2

3.5 Other Strategies for Integration

Calculus Volume 23.5 Other Strategies for Integration

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Table of contents
  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Learning Objectives

  • 3.5.1 Use a table of integrals to solve integration problems.
  • 3.5.2 Use a computer algebra system (CAS) to solve integration problems.

In addition to the techniques of integration we have already seen, several other tools are widely available to assist with the process of integration. Among these tools are integration tables, which are readily available in many books, including the appendices to this one. Also widely available are computer algebra systems (CAS), which are found on calculators and in many campus computer labs, and are free online.

Tables of Integrals

Integration tables, if used in the right manner, can be a handy way either to evaluate or check an integral quickly. Keep in mind that when using a table to check an answer, it is possible for two completely correct solutions to look very different. For example, in Trigonometric Substitution, we found that, by using the substitution x=tanθ,x=tanθ, we can arrive at

dx1+x2=ln(x+x2+1)+C.dx1+x2=ln(x+x2+1)+C.

However, using x=sinhθ,x=sinhθ, we obtained a different solution—namely,

dx1+x2=sinh−1x+C.dx1+x2=sinh−1x+C.

We later showed algebraically that the two solutions are equivalent. That is, we showed that sinh−1x=ln(x+x2+1).sinh−1x=ln(x+x2+1). In this case, the two antiderivatives that we found were actually equal. This need not be the case. However, as long as the difference in the two antiderivatives is a constant, they are equivalent.

Example 3.36

Using a Formula from a Table to Evaluate an Integral

Use the table formula

a2u2u2du=a2u2usin−1ua+Ca2u2u2du=a2u2usin−1ua+C

to evaluate 16e2xexdx.16e2xexdx.

Computer Algebra Systems

If available, a CAS is a faster alternative to a table for solving an integration problem. Many such systems are widely available and are, in general, quite easy to use.

Example 3.37

Using a Computer Algebra System to Evaluate an Integral

Use a computer algebra system to evaluate dxx24.dxx24. Compare this result with ln|x242+x2|+C,ln|x242+x2|+C, a result we might have obtained if we had used trigonometric substitution.

Media

You can access an integral calculator for more examples.

Example 3.38

Using a CAS to Evaluate an Integral

Evaluate sin3xdxsin3xdx using a CAS. Compare the result to 13cos3xcosx+C,13cos3xcosx+C, the result we might have obtained using the technique for integrating odd powers of sinxsinx discussed earlier in this chapter.

Checkpoint 3.21

Use a CAS to evaluate dxx2+4.dxx2+4.

Section 3.5 Exercises

Use a table of integrals to evaluate the following integrals.

244.

0 4 x 1 + 2 x d x 0 4 x 1 + 2 x d x

245.

x + 3 x 2 + 2 x + 2 d x x + 3 x 2 + 2 x + 2 d x

246.

x 3 1 + 2 x 2 d x x 3 1 + 2 x 2 d x

247.

1 x 2 + 6 x d x 1 x 2 + 6 x d x

248.

x x + 1 d x x x + 1 d x

249.

x · 2 x 2 d x x · 2 x 2 d x

250.

1 4 x 2 + 25 d x 1 4 x 2 + 25 d x

251.

d y 4 y 2 d y 4 y 2

252.

sin 3 ( 2 x ) cos ( 2 x ) d x sin 3 ( 2 x ) cos ( 2 x ) d x

253.

csc ( 2 w ) cot ( 2 w ) d w csc ( 2 w ) cot ( 2 w ) d w

254.

2 y d y 2 y d y

255.

0 1 3 x d x x 2 + 8 0 1 3 x d x x 2 + 8

256.

−1 / 4 1 / 4 sec 2 ( π x ) tan ( π x ) d x −1 / 4 1 / 4 sec 2 ( π x ) tan ( π x ) d x

257.

0 π / 2 tan 2 ( x 2 ) d x 0 π / 2 tan 2 ( x 2 ) d x

258.

cos 3 x d x cos 3 x d x

259.

tan 5 ( 3 x ) d x tan 5 ( 3 x ) d x

260.

sin 2 y cos 3 y d y sin 2 y cos 3 y d y

Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.

261.

[T] dw1+sec(w2)dw1+sec(w2)

262.

[T] dw1cos(7w)dw1cos(7w)

263.

[T] 0tdt4cost+3sint0tdt4cost+3sint

264.

[T] x293xdxx293xdx

265.

[T] dxx1/2+x1/3dxx1/2+x1/3

266.

[T] dxxx1dxxx1

267.

[T] x3sinxdxx3sinxdx

268.

[T] xx49dxxx49dx

269.

[T] x1+ex2dxx1+ex2dx

270.

[T] 35x2xdx35x2xdx

271.

[T] dxxx1dxxx1

272.

[T] excos−1(ex)dxexcos−1(ex)dx

Use a calculator or CAS to evaluate the following integrals.

273.

[T] 0π/4cos(2x)dx0π/4cos(2x)dx

274.

[T] 01x·ex2dx01x·ex2dx

275.

[T] 082xx2+36dx082xx2+36dx

276.

[T] 02/314+9x2dx02/314+9x2dx

277.

[T] dxx2+4x+13dxx2+4x+13

278.

[T] dx1+sinxdx1+sinx

Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table.

279.

d x x 2 + 2 x + 10 d x x 2 + 2 x + 10

280.

d x x 2 6 x d x x 2 6 x

281.

e x e 2 x 4 d x e x e 2 x 4 d x

282.

cos x sin 2 x + 2 sin x d x cos x sin 2 x + 2 sin x d x

283.

arctan ( x 3 ) x 4 d x arctan ( x 3 ) x 4 d x

284.

ln | x | arcsin ( ln | x | ) x d x ln | x | arcsin ( ln | x | ) x d x

Use tables to perform the integration.

285.

d x x 2 + 16 d x x 2 + 16

286.

3 x 2 x + 7 d x 3 x 2 x + 7 d x

287.

d x 1 cos ( 4 x ) d x 1 cos ( 4 x )

288.

d x 4 x + 1 d x 4 x + 1

289.

Find the area bounded by y(4+25x2)=5,x=0,y=0,andx=4.y(4+25x2)=5,x=0,y=0,andx=4. Use a table of integrals or a CAS.

290.

The region bounded between the curve y=11+cosx,0.3x1.1,y=11+cosx,0.3x1.1, and the x-axis is revolved about the x-axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)

291.

Use substitution and a table of integrals to find the area of the surface generated by revolving the curve y=ex,0x3,y=ex,0x3, about the x-axis. (Round the answer to two decimal places.)

292.

[T] Use an integral table and a calculator to find the area of the surface generated by revolving the curve y=x22,0x1,y=x22,0x1, about the x-axis. (Round the answer to two decimal places.)

293.

[T] Use a CAS or tables to find the area of the surface generated by revolving the curve y=cosx,0xπ2,y=cosx,0xπ2, about the x-axis. (Round the answer to two decimal places.)

294.

Find the length of the curve y=x24y=x24 over [0,8].[0,8].

295.

Find the length of the curve y=exy=ex over [0,ln(2)].[0,ln(2)].

296.

Find the area of the surface formed by revolving the graph of y=2xy=2x over the interval [0,9][0,9] about the x-axis.

297.

Find the average value of the function f(x)=1x2+1f(x)=1x2+1 over the interval [−3,3].[−3,3].

298.

Approximate the arc length of the curve y=tan(πx)y=tan(πx) over the interval [0,14].[0,14]. (Round the answer to three decimal places.)

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