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2.6E: Exercises

This page is a draft and is under active development. 

( \newcommand{\kernel}{\mathrm{null}\,}\)

Improper Integrals

Evaluate the following integrals. If the integral is not convergent, answer “divergent.”

Exercise 2.6E.1

42dx(x3)2

Answer

divergent

Exercise 2.6E.2

014+x2dx

Answer

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Exercise 2.6E.3

2014x2dx

Answer

π2

Exercise 2.6E.4

11xlnxdx

Answer

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Exercise 2.6E.5

1xexdx

Answer

2e

Exercise 2.6E.6

xx2+1dx

Answer

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Exercise 2.6E.7

Without integrating, determine whether the integral 11x3+1dx converges or diverges by comparing the function f(x)=1x3+1 with g(x)=1x3.

Answer

Converges

Exercise 2.6E.8

Without integrating, determine whether the integral 11x+1dx converges or diverges.

Answer

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Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.

Exercise 2.6E.9

0excosxdx

Answer

Converges to 1/2.

Exercise 2.6E.10

1lnxxdx

Answer

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Exercise 2.6E.11

10lnxxdx

Answer

4

Exercise 2.6E.12

10lnxdx

Answer

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Exercise 2.6E.13

1x2+1dx

Answer

π

Exercise 2.6E.14

51dxx1

Answer

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Exercise 2.6E.15

22dx(1+x)2

Answer

diverges

Exercise 2.6E.16

0exdx

Answer

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Exercise 2.6E.17

0sinxdx

Answer

diverges

Exercise 2.6E.18

ex1+e2xdx

Answer

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Exercise 2.6E.19

10dx3x

Answer

1.5

Exercise 2.6E.20

20dxx3

Answer

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Exercise 2.6E.21

21dxx3

Answer

diverges

Exercise 2.6E.22

10dx1x2

Answer

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Exercise 2.6E.23

301x1dx

Answer

diverges

Exercise 2.6E.24

15x3dx

Answer

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Exercise 2.6E.25

535(x4)2dx

Answer

diverges

Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.

Exercise 2.6E.26

1dxx2+4x; compare with 1dxx2.

Answer

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Exercise 2.6E.27

1dxx+1; compare with 1dx2x.

Answer

Both integrals diverge.

Evaluate the integrals. If the integral diverges, answer “diverges.”

Exercise 2.6E.28

1dxxe

Answer

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Exercise 2.6E.29

10dxxπ

Answer

diverges

Exercise 2.6E.30

10dx1x

Answer

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Exercise 2.6E.31

10dx1x

Answer

diverges

Exercise 2.6E.32

0dxx2+1

Answer

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Exercise 2.6E.33

11dx1x2

Answer

π

Exercise 2.6E.34

10lnxxdx

Answer

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Exercise 2.6E.35

e0ln(x)dx

Answer

0.0

Exercise 2.6E.36

0xexdx

Answer

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Exercise 2.6E.37

x(x2+1)2dx

Answer

0.0

Exercise 2.6E.38

0exdx

Answer

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Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.

Exercise 2.6E.39

90dx9x

Answer

6.0

Exercise 2.6E.40

127dxx2/3

Answer

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Exercise 2.6E.41

30dx9x2

Answer

π2

Exercise 2.6E.42

246dttt236

Answer

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Exercise 2.6E.43

40xln(4x)dx

Answer

8ln(16)4

Exercise 2.6E.44

30x9x2dx

Answer

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Exercise 2.6E.45

Evaluate t.5dx1x2. (Be careful!) (Express your answer using three decimal places.)

Answer

1.047

Exercise 2.6E.46

Evaluate 41dxx21. (Express the answer in exact form.)

Answer

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Exercise 2.6E.47

Evaluate 2dx(x21)3/2.

Answer

1+23

Exercise 2.6E.48

Find the area of the region in the first quadrant between the curve y=e6x and the x-axis.

Answer

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Exercise 2.6E.49

Find the area of the region bounded by the curve y=7x2, the x-axis, and on the left by x=1.

Answer

7.0

Exercise 2.6E.50

Find the area under the curve y=1(x+1)3/2, bounded on the left by x=3.

Answer

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Exercise 2.6E.51

Find the area under y=51+x2 in the first quadrant.

Answer

5π2

Exercise 2.6E.52

Find the volume of the solid generated by revolving about the x-axis the region under the curve y=3x from x=1 to x=.

Answer

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Exercise 2.6E.53

Find the volume of the solid generated by revolving about the y-axis the region under the curve y=6e2x in the first quadrant.

Answer

3π

Exercise 2.6E.54

Find the volume of the solid generated by revolving about the x-axis the area under the curve y=3ex in the first quadrant.

Answer

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The Laplace transform of a continuous function over the interval [0,) is defined by F(s)=0esxf(x)dx (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F.

Exercise 2.6E.55

f(x)=1

Answer

1s,s>0

Exercise 2.6E.56

f(x)=x

Answer

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Exercise 2.6E.57

f(x)=cos(2x)

Answer

ss2+4,s>0

Exercise 2.6E.58

f(x)=eax

Answer

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Exercise 2.6E.59

Use the formula for arc length to show that the circumference of the circle x2+y2=1 is 2π.

Answer

Answers will vary.

A function is a probability density function if it satisfies the following definition: f(t)dt=1. The probability that a random variable x lies between a and b is given by P(axb)=baf(t)dt.

Exercise 2.6E.60

Show that f(x)={0ifx<07e7xifx0 is a probability density function.

Answer

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Exercise 2.6E.61

Find the probability that x is between 0 and 0.3. (Use the function defined in the preceding problem.) Use four-place decimal accuracy.

Answer

0.8775


2.6E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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