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Chapter 7

Techniques of Integration

Educators


Problem 1

Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{2}} \cos 5x \cos 2x\ dx $ ; entry 80

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Problem 2

Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral.

$ \displaystyle \int_0^1 \sqrt{x - x^2}\ dx $ ; entry 113

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Problem 3

Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral.

$ \displaystyle \int_1^2 \sqrt{4x^2 - 3}\ dx $ ; entry 39

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Problem 4

Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral.

$ \displaystyle \int_0^1 \tan^3 \left (\frac{\pi x}{6} \right)\ dx $ ; entry 69

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Problem 5

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{8}} \arctan 2x\ dx $

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Problem 6

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^2 x^2 \sqrt{4 - x^2}\ dx $

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Problem 7

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\cos x}{\sin^2 x - 9}\ dx $

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Problem 8

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{e^x}{4 - e^{2x}}\ dx $

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Problem 9

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\sqrt{9x^2 + 4}}{x^2}\ dx $

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Problem 10

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\sqrt{2y^2 - 3}}{y^2}\ dy $

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Problem 11

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^\pi \cos^6 \theta\ d \theta $

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Problem 12

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int x \sqrt{2 + x^4}\ dx $

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Problem 13

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\arctan \sqrt{x}}{\sqrt{x}}\ dx $

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Problem 14

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^{\pi} x^3 \sin x\ dx $

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Problem 15

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\coth \frac{1}{y}}{y^2}\ dy $

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Problem 16

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{e^{3t}}{\sqrt{e^{2t} - 1}}\ dt $

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Problem 17

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int y \sqrt{6 + 4y -4y^2}\ dy $

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Problem 18

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{dx}{2x^3 - 3x^2} $

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Problem 19

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \sin^2 x \cos x \ln (\sin x)\ dx $

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Problem 20

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\sin 2 \theta}{\sqrt{5 - \sin \theta}}\ d \theta $

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Problem 21

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{e^x}{3 - e^{2x}}\ dx $

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Problem 22

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^2 x^3 \sqrt{4x^2 - x^4}\ dx $

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Problem 23

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \sec^5 x\ dx $

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Problem 24

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int x^3 \arcsin (x^2)\ dx $

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Problem 25

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\sqrt{4 + (\ln x)^2}}{x}\ dx $

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Problem 26

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^1 x^4 e^{-x}\ dx $

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Problem 27

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\cos^{-1} (x^{-2})}{x^3}\ dx $

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Problem 28

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{dx}{\sqrt{1 - e^{2x}}} $

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Problem 29

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \sqrt{e^{2x} - 1}\ dx $

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Problem 30

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int e^t \sin (\alpha t - 3)\ dt $

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Problem 31

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{x^4\ dx}{\sqrt{x^{10} - 2}} $

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Problem 32

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int \frac{\sec^2 \theta \tan^2 \theta}{\sqrt{9 - \tan^2 \theta}}\ d \theta $

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Problem 33

The region under the curve $ y = \sin^2 x $ from 0 to $ \pi $ is rotated about the x-axis. Find the volume of the resulting solid.

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Problem 34

Find the volume of the solid obtained when the region under the curve $ y = \arcsin x $, $ x \ge 0 $, is rotated about the y-axis.

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Problem 35

Verify Formula 53 in the Table of Integrals (a) by differentiation and (b) by using the substitution $ t = a + bu $.

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Problem 36

Verify Formula 31 (a) by differentiation and (b) by substituting $ u = a \sin \theta $.

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Problem 37

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int \sec^4 x\ dx $

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Problem 38

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int \csc^5 x\ dx $

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Problem 39

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int x^2 \sqrt{x^2 + 4}\ dx $

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Problem 40

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int \frac{dx}{e^x (3e^x + 2)} $

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Problem 41

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int \cos^4 x\ dx $

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Problem 42

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int x^2 \sqrt{1 - x^2}\ dx $

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Problem 43

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int \tan^5 x\ dx $

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Problem 44

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.

$ \displaystyle \int \frac{1}{\sqrt{1 + \sqrt[3]{x}}}\ dx $

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Problem 45

Use the table of integrals to evaluate $ F(x) = \int f(x)\ dx $, where
$$ f(x) = \frac{1}{x \sqrt{1 - x^2}} $$
What is the domain of $ f $ and $ F $?
(b) Use a $ CAS $ to evaluate $ F(x) $. What is the domain of the function $ F $ that the $ CAS $ produces? Is there a discrepancy between this domain and the domain of the function $ F $ that you found in part (a)?

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Problem 46

Computer algebra systems sometimes need a helping hand from human beings. Try to evaluate
$$ \int (1 + \ln x) \sqrt{1 + (x \ln x)^2}\ dx $$
with a computer algebra system. If it doesn't return an answer, make a substitution that changes the integral into one that the $ CAS $ can evaluate.

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