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Mathematics LibreTexts

2.E: Chapter Review Exercises

This page is a draft and is under active development. 

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Exercise 2.E.1

2xln(x)dx

Answer

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Exercise 2.E.2

3sin3(x)cos3(x)dx

Answer

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Exercise 2.E.3

(4x2+x+4x3+xdx

Answer

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For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

Exercise 2.E.4

exsin(x)dx cannot be integrated by parts.

Answer

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Exercise 2.E.5

1x4+1dx cannot be integrated using partial fractions.

Answer

False

Exercise 2.E.6

In numerical integration, increasing the number of points decreases the error.

Answer

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Exercise 2.E.7

Integration by parts can always yield the integral.

Answer

False

For the following exercises, evaluate the integral using the specified method.

Exercise 2.E.8

x2sin(4x)dx using integration by parts

Answer

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Exercise 2.E.9

1x2x2+16dx using trigonometric substitution

Answer

x2+1616x+C

Exercise 2.E.10

xln(x)dx using integration by parts

Answer

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Exercise 2.E.11

3xx3+2x25x6dx using partial fractions

Answer

110(4ln(2x)+5ln(x+1)9ln(x+3))+C

Exercise 2.E.12

x5(4x2+4)5/2dx using trigonometric substitution

Answer

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Exercise 2.E.13

4sin2(x)sin2(x)cos(x)dx using a table of integrals or a CAS

Answer

4sin2(x)sin(x)x2+C

For the following exercises, integrate using whatever method you choose.

Exercise 2.E.14

sin2(x)cos2(x)dx

Answer

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Exercise 2.E.15

x3x2+2dx

Answer

115(x2+2)3/2(3x24)+C

Exercise 2.E.16

3x2+1x42x3x2+2xdx

Answer

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Exercise 2.E.17

1x4+4dx

Answer

116ln(x2+2x+2x22x+2)18tan1(1x)+18tan1(x+1)+C

Exercise 2.E.18

3+16x4x4dx

Answer

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For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.

Exercise 2.E.19

21x5+2dx

Answer

M4=3.312,T4=3.354,S4=3.326

Exercise 2.E.20

π0esin(x2)dx

Answer

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Exercise 2.E.21

41ln(1/x)xdx

Answer

M4=0.982,T4=0.917,S4=0.952

For the following exercises, evaluate the integrals, if possible.

Exercise 2.E.22

11xndx, for what values of n does this integral converge or diverge?

Answer

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Exercise 2.E.23

1exxdx

Answer

approximately 0.2194

For the following exercises, consider the gamma function given by Γ(a)=0eyya1dy.

Exercise 2.E.24

Show that Γ(a)=(a1)Γ(a1).

Answer

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Exercise 2.E.25

Extend to show that Γ(a)=(a1)!, assuming a is a positive integer.

Answer

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The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km/h. The graph represents its velocity.

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Exercise 2.E.26

Use the graph to estimate the velocity every 20 sec and fit to a graph of the form v(t)=aexpbxsin(cx)+d. (Hint: Consider the time units.)

Answer

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Exercise 2.E.27

Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.

Answer

Answers may vary. Ex: 9.405 km


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

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