2.E: Chapter Review Exercises
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Exercise 2.E.1
∫2xln(x)dx
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Exercise 2.E.2
∫3sin3(x)cos3(x)dx
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Exercise 2.E.3
∫(4x2+x+4x3+xdx
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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.
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Exercise 2.E.5
∫1x4+1dx cannot be integrated using partial fractions.
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False
Exercise 2.E.6
In numerical integration, increasing the number of points decreases the error.
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Exercise 2.E.7
Integration by parts can always yield the integral.
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False
For the following exercises, evaluate the integral using the specified method.
Exercise 2.E.8
∫x2sin(4x)dx using integration by parts
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Exercise 2.E.9
∫1x2√x2+16dx using trigonometric substitution
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−√x2+1616x+C
Exercise 2.E.10
∫√xln(x)dx using integration by parts
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Exercise 2.E.11
∫3xx3+2x2−5x−6dx using partial fractions
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110(4ln(2−x)+5ln(x+1)−9ln(x+3))+C
Exercise 2.E.12
∫x5(4x2+4)5/2dx using trigonometric substitution
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Exercise 2.E.13
∫√4−sin2(x)sin2(x)cos(x)dx using a table of integrals or a CAS
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−√4−sin2(x)sin(x)−x2+C
For the following exercises, integrate using whatever method you choose.
Exercise 2.E.14
∫sin2(x)cos2(x)dx
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Exercise 2.E.15
∫x3√x2+2dx
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115(x2+2)3/2(3x2−4)+C
Exercise 2.E.16
∫3x2+1x4−2x3−x2+2xdx
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Exercise 2.E.17
∫1x4+4dx
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116ln(x2+2x+2x2−2x+2)−18tan−1(1−x)+18tan−1(x+1)+C
Exercise 2.E.18
∫√3+16x4x4dx
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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
∫21√x5+2dx
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M4=3.312,T4=3.354,S4=3.326
Exercise 2.E.20
∫√π0e−sin(x2)dx
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Exercise 2.E.21
∫41ln(1/x)xdx
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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?
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Exercise 2.E.23
∫∞1e−xxdx
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approximately 0.2194
For the following exercises, consider the gamma function given by Γ(a)=∫∞0e−yya−1dy.
Exercise 2.E.24
Show that Γ(a)=(a−1)Γ(a−1).
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Exercise 2.E.25
Extend to show that Γ(a)=(a−1)!, assuming a is a positive integer.
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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.
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.)
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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.
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Answers may vary. Ex: 9.405 km