Deck 13: Techniques of Integration

Question 1

Identify u and dv for finding the integral using integration by parts. $\int x ^ { 4 } e ^ { 7 x } d x$

A) $u = x ^ { 4 } ; d v = e ^ { 7 x } d x$
B) $u = \int x ^ { 4 } ; d v = \int e ^ { 7 x } d x$
C) $u = \int x ^ { 4 } d x , d v = e ^ { 7 x } d x$
D) $u = \int x ^ { 4 } d x ; d v = \int e ^ { 7 x } d x$
E) $u = x ^ { 4 } d x , d v = \int e ^ { 7 x } d x$

Answer 1

$u = x ^ { 4 } ; d v = e ^ { 7 x } d x$

Question 2

Use integration by parts to evaluate $\int 4 x e ^ { - 5 x } d x$

A) $- \frac { ( 5 x + 1 ) \left( 4 e ^ { - 5 x } \right) } { 25 } + C$
B) $- \frac { e ^ { - 5 x } } { 5 } + C$
C) $- \frac { \left( 4 e ^ { - 5 x } \right) } { 25 } + C$
D) $- \frac { x \left( 4 e ^ { - 5 x } \right) } { 5 } + C$
E) $\frac { e ^ { - 5 x } } { 5 } + C$

Answer 2

$- \frac { ( 5 x + 1 ) \left( 4 e ^ { - 5 x } \right) } { 25 } + C$

Question 3

Find the indefinite integral. $\int \frac { t } { \sqrt { 1 + 7 t } } d t$

A) $\frac { ( 7 t - 2 ) \sqrt { 7 t + 1 } } { 49 } + C$
B) $\frac { 2 ( 7 t - 2 ) \sqrt { 7 t + 1 } } { 147 } + C$
C) $\frac { 2 ( 2 - 7 t ) \sqrt { 7 t + 1 } } { 147 } + C$
D) $\frac { 2 ( 7 t - 2 ) \sqrt { 7 t + 1 } } { 21 } + C$
E)none of the above

Answer 3

$\frac { 2 ( 7 t - 2 ) \sqrt { 7 t + 1 } } { 147 } + C$

Question 4

Use integration by parts to find the integral below. $\int 5 x ^ { n } \ln a x d x ( a \neq 0 , n \neq - 1 )$

A) $\frac { 5 x ^ { n } } { n } \ln a x - \frac { 5 } { n ^ { 2 } } x ^ { n } + C$
B) $\frac { 6 x ^ { n + 1 } } { n + 1 } \ln a x - \frac { 6 } { ( n + 1 ) ^ { 2 } } x ^ { n + 1 } + C$
C) $\frac { 5 x ^ { n } } { n } - \frac { 5 } { ( n + 1 ) ^ { 2 } } \ln a x + C$
D) $\frac { 5 x ^ { n + 1 } } { n + 1 } \ln a x - \frac { 5 } { ( n + 1 ) ^ { 2 } } x ^ { n + 1 } + C$
E) $\frac { 6 x ^ { n + 1 } } { n + 1 } - \frac { 6 } { n ^ { 2 } } \ln a x + C$

Question 5

A model for the ability M of a child to memorize, measured on a scale from 0 to 10, is $M = 1 + 1.6 t \ln t , 0 < t \leq 3$ where t is the child's age in years. Find the average value predicted by the model for a child's ability to memorize between second and third birthdays. Round your answer to three decimal places.

A)5.992
B)4.892
C)6.792
D)3.992
E)4.692

Question 6

Find the integral below using an integral table. $\int \frac { 6 } { 64 - x ^ { 2 } } d x$

A) $\frac { 3 } { 8 } \ln \left| \frac { 8 + x } { 8 - x } \right| + C$
B) $\frac { 3 } { 8 } \ln \left| \frac { 64 + x } { 64 - x } \right| + C$
C) $\frac { 3 } { 8 } \ln \left| \frac { 64 - x } { 64 + x } \right| + C$
D) $\frac { 1 } { 8 } \ln \left| \frac { 8 - x } { 8 + x } \right| + C$
E) $\frac { 3 } { 8 } \ln \left| \frac { 8 - x } { 8 + x } \right| + C$

Question 7

Find the indefinite integral. $\int \frac { \ln v } { 9 v ^ { 3 } } d v$

A) $- \frac { 1 } { 18 v ^ { 2 } } ( 2 \ln v + 1 ) + C$
B) $\left. \frac { 1 } { 18 v ^ { 2 } } ( 2 \ln v + 1 ) + C \right)$
C) $\frac { 1 } { 36 v ^ { 2 } } ( 2 \ln v - 1 ) + C$
D) $- \frac { 1 } { 36 v ^ { 2 } } ( 2 \ln v - 1 ) + C$
E) $- \frac { 1 } { 36 v ^ { 2 } } ( 2 \ln v + 1 ) + C$

Question 8

Present Value of a Continuous Stream of Income. An electronics company generates a continuous stream of income of $4 t$ million dollars per year, where t is the number of years that the company has been in operation. Find the present value of this stream of income over the first 9 years at a continuous interest rate of 12%. Round answer to one decimal place.

A)$143.7 million
B)$81.6 million
C)$182.7 million
D)$343.2 million
E)$85.8 million

Question 9

Find the indefinite integral. $\int v \ln ( v + 2 ) d v$

A) $\left( \frac { v ^ { 2 } - 4 } { 2 } \right) \ln ( v + 2 ) - \frac { v ^ { 2 } - 4 v } { 4 } + C$
B) $\left( \frac { v ^ { 2 } - 4 } { 2 } \right) \ln ( v + 2 ) + \frac { v ^ { 2 } - 4 v } { 4 } + C$
C) $\left( \frac { v ^ { 2 } + 4 } { 2 } \right) \ln ( v + 2 ) - \frac { v ^ { 2 } + 2 v } { 4 } + C$
D) $\left( \frac { v ^ { 2 } - 4 } { 2 } \right) \ln ( v + 2 ) - \frac { v ^ { 2 } - 4 v } { 2 } + C$
E) $\left( \frac { v ^ { 2 } - 4 } { 2 } \right) \ln ( v + 2 ) + \frac { v ^ { 2 } - 2 v } { 4 } + C$

Question 10

Use integration by parts to evaluate $\int 3 x ^ { 3 } \ln x d x$ .

A) $\frac { - 3 x ^ { 4 } ( 4 \ln ( x ) + 1 ) } { 4 } + C$
B) $\frac { 3 x ^ { 3 } ( 3 \ln ( x ) + 1 ) } { 3 } + C$
C) $\frac { 3 x ^ { 4 } ( 4 \ln ( x ) + 1 ) } { 16 } + C$
D) $\frac { 3 x ^ { 3 } ( 3 \ln ( x ) - 1 ) } { 9 } + C$
E) $\frac { 3 x ^ { 4 } ( 4 \ln ( x ) - 1 ) } { 16 } + C$

Question 11

Use a table of integrals with forms involving $e ^ { u }$ to find the integral. $\int \frac { - 7 } { 1 + e ^ { - 6 x } } d x$

A) $7 x - \frac { 7 } { 6 } \ln \left( 1 - e ^ { - 6 x } \right) + C$
B) $7 x - \frac { 7 } { 6 } \ln \left( 1 + e ^ { - 6 x } \right) + C$
C) $- 7 x + \frac { 7 } { 6 } \ln \left( 1 + e ^ { - 6 x } \right) + C$
D) $- 7 x + \frac { 7 } { 6 } \ln \left( 1 - e ^ { - 6 x } \right) + C$
E) $- 7 x - \frac { 7 } { 6 } \ln \left( 1 + e ^ { - 6 x } \right) + C$

Question 12

Evaluate the definite integral $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { 2 x } d x$ . Round your answer to three decimal places.

A)4.195
B)9.486
C)1.597
D)8.986
E)0.473

Question 13

Use integration by parts to evaluate $\int x ^ { 2 } e ^ { - 3 x } d x$ . Note that evaluation may require integration by parts more than once.

A) $- \frac { \left( 2 + 6 x + 9 x ^ { 2 } \right) e ^ { - 3 x } } { 27 } + C$
B) $\frac { \left( 1 + 6 x - 3 x ^ { 2 } \right) e ^ { - 3 x } } { 9 } + C$
C) $- \frac { \left( 2 + 9 x + 9 x ^ { 2 } \right) e ^ { - 3 x } } { 27 } + C$
D) $\frac { \left( 2 - 3 x + 9 x ^ { 2 } \right) e ^ { - 3 x } } { 9 } + C$
E) $- \frac { \left( 1 - 3 x + 9 x ^ { 2 } \right) e ^ { - 3 x } } { - 27 } + C$

Question 14

Use a table of integrals to find the indefinite integral $\int ( \ln 3 x ) ^ { 2 } d x$ .

A) $x \left[ 2 - 2 \ln ( 3 x ) + ( \ln 3 x ) ^ { 2 } \right] + C$
B) $x \left[ 2 \ln ( 3 x ) + ( \ln 3 x ) ^ { 2 } \right] + C$
C) $3 x \left[ 2 + 2 \ln ( 3 x ) + ( \ln 3 x ) ^ { 2 } \right] + C$
D) $x [ 2 - 2 \ln ( 3 x ) - ( \ln 3 x ) ] ^ { 2 } + C$
E) $3 x [ 2 \ln ( 3 x ) + ( \ln 3 x ) ] ^ { 2 } + C$

Question 15

Find the indefinite integral. $\int t \sqrt { 4 t + 9 } d t$

A) $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t + 3 ) } { 20 } + C$
B) $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t - 6 ) } { 20 } + C$
C) $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t - 3 ) } { 20 } + C$
D) $\frac { ( 4 t - 9 ) ^ { 3 / 2 } ( 2 t + 3 ) } { 20 } + C$
E) $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t + 6 ) } { 20 } + C$

Question 16

Find the definite integral. $\int _ { 1 } ^ { 4 } x ^ { 3 } \ln x d x$

A) $\frac { 2048 \ln 4 - 255 } { 16 }$
B) $\frac { 1024 \ln 4 - 255 } { 16 }$
C) $\frac { - 1024 \ln 4 - 255 } { 16 }$
D) $\frac { 1024 \ln 4 + 255 } { 16 }$
E)none of the above

Question 17

Find the indefinite integral. $\int \frac { 6 ( \ln x ) ^ { 2 } } { x ^ { 2 } } d x$

A) $\frac { 6 \left( ( \ln x ) ^ { 2 } - 2 \ln x + 2 \right) } { x } + C$
B) $- \frac { 6 \left( ( \ln x ) ^ { 2 } + 2 \ln x + 2 \right) } { x } + C$
C) $- 6 \left( ( \ln x ) ^ { 2 } + 2 \ln x + 2 \right) + C$
D) $- \frac { 6 \left( 2 ( \ln x ) ^ { 2 } + 2 \right) } { x } + C$
E) $\frac { 6 \left( 2 ( \ln x ) ^ { 2 } + 2 \right) } { x } + C$

Question 18

Find the indefinite integral. $\int \frac { 3 x ^ { 2 } } { e ^ { x } } d x$

A) $3 \left( x ^ { 2 } - 2 x + 2 \right) e ^ { - x } + C$
B) $- \left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$
C) $\left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$
D) $- 3 x \left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$
E) $- 3 \left( x ^ { 2 } + 2 x + 2 \right) e ^ { - x } + C$

Question 19

Use integration by parts to find the integral below. $\int \ln x ^ { 3 } d x$

A) $\ln x ^ { 4 } - 3 x ^ { 4 } + C$
B) $x \ln x ^ { 4 } - 4 x ^ { 4 } + C$
C) $\ln x ^ { 4 } - x ^ { 4 } + C$
D) $x \ln x ^ { 3 } - 3 x + C$
E) $4 x \ln x ^ { 4 } - 4 x + C$

Question 20

Identify u and dv for finding the integral using integration by parts. $\int x ^ { 3 } \ln 9 x$ dx

A) $u = \int \ln 9 x$ dx; $d v = \int x ^ { 3 }$ dx
B) $u = \ln 9 x , d v = x ^ { 3 } d x$
C) $u = \int \ln 9 x , d v = x ^ { 3 } d x$
D) $u = \ln 9 x , d v = \int x ^ { 3 } d x$
E) $u = \int \ln 9 x , d v = \int x ^ { 3 } d x$

Question 21

Approximate the value of the definite integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of n. Round your answers to three significant digits. $\int _ { 0 } ^ { 2 } e ^ { - x ^ { 2 } } d x , n = 4$

A)a. Trapezoidal Rule: $\approx 1.881$ b. Simpson's Rule: $\approx 0.882$
B)a. Trapezoidal Rule: $\approx 0.881$ b. Simpson's Rule: $\approx 0.882$
C)a. Trapezoidal Rule: $\approx 0.881$ b. Simpson's Rule: $\approx 1.882$
D)a. Trapezoidal Rule: $\approx 0.081$ b. Simpson's Rule: $\approx 0.882$
E)a. Trapezoidal Rule: $\approx 0.881$ b. Simpson's Rule: $\approx 0.082$

Question 22

The probability of recall in an experiment is modeled by $P ( a \leq x \leq b ) = \int _ { a } ^ { b } \frac { 75 } { 14 } \left( \frac { x } { \sqrt { 4 + 5 x } } \right) d x , 0 \leq x \leq 1$ where x is the percent of recall. What is the probability of recalling between 50% and 70%? Round your answer to three decimal places.

A)0.243
B)0.206
C)0.650
D)0.163
E)0.832

Question 23

Evaluate the definite integral $\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x$ dx. Round your answer to three decimal places.

A)7.499
B)8.562
C)5.896
D)6.999
E)6.236

Question 24

Approximate the definite integral "by hand," using the Trapezoidal Rule with $n = 4$ trapezoids. Round answer to three decimal places. $\int _1^ { 2 } \frac { 7 } { x } d x$

A)19.517
B)6.192
C)3.096
D)4.879
E)24.767

Question 25

The rate of change in the number of subscribers $S$ to a newly introduced magazine is modeled by $\frac { d S } { d t } = 1000 t ^ { 2 } e ^ { - 1 } , 0 \leq t \leq 6$ where $t$ is the time in years. Use Simpson's Rule $n = 12$ with to estimate the total increase in the number of subscribers during the first 6 years.

A) $\approx$ 1870 subscribers
B) $\approx$ 1780 subscribers
C) $\approx$ 1800 subscribers
D) $\approx$ 1878 subscribers
E) $\approx$ 1987 subscribers

Question 26

Use Simpson's Rule to approximate the revenue for the marginal revenue function $\frac { d R } { d x } = 5 \sqrt { 8000 - x ^ { 3 } }$ with n = 4. Assume that the number of units sold, x, increases from 14 to 18. Round your answer to one decimal place.

A)$1439.03
B)$1346.14
C)$1602.40
D)$1230.54
E)$678.36

Question 27

Use the error formulas to find n such that the error in the approximation of the definite integral $\int _ { 3 } ^ { 6 } \frac { 1 } { x } d x$ is less than 0.0001 using the Trapezoidal Rule.

A)43
B)44
C)42
D)40
E)41

Question 28

Use the Trapezoidal Rule to approximate the value of the definite integral $\int _ { 0 } ^ { 3 } \sqrt { 1 + x } d x , n = 4$ . Round your answer to three decimal places.

A)2.7931
B)2.7955
C)4.6552
D)4.6615
E)6.7643

Question 29

Evaluate the definite integral $\int _ { 2 } ^ { 6 } \sqrt { 7 + x ^ { 2 } } d x$ . Round your answer to three decimal places.

A)31.060
B)25.997
C)37.693
D)34.376
E)19.364

Question 30

Use a table of integrals with forms involving a + bu to find $\int \frac { x ^ { 2 } } { 8 + 11 x } d x$

A) $\frac { 1 } { 121 } ( 11 x - 8 \ln | 8 + 11 x | ) + C$
B) $\frac { 1 } { 1331 } \left( 11 x - \frac { 64 } { 8 + 11 x } - 16 \ln | 8 + 11 x | \right) + C$
C) $\frac { 1 } { 1331 } \left( \frac { 11 x } { 2 } ( 11 x - 16 ) + 64 \ln | 8 + 11 x | \right) + C$
D) $\frac { 1 } { 121 } \left( \frac { 11 x } { 2 } ( 11 x - 16 ) + 64 \ln | 8 + 11 x | \right) + C$
E) $\frac { 1 } { 121 } \left( 11 x - \frac { 64 } { 8 + 11 x } - 16 \ln | 8 + 11 x | \right) + C$

Question 31

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $\int _ { 0 } ^ { 1 } \left( \frac { x ^ { 2 } } { 2 } + 1 \right) d x , n = 4$

A)a. Exact: $\approx 1.9667$ b. Trapezoidal Rule: $\approx 1.1719$ c. Simpson's Rule: $\approx 1.1667$
B)a. Exact: $\approx 1.1667$ b. Trapezoidal Rule: $\approx 1.1719$ c. Simpson's Rule: $\approx 1.9667$
C)a. Exact: $\approx 1.1667$ b. Trapezoidal Rule: $\approx 1.9719$ c. Simpson's Rule: $\approx 1.1667$
D)a. Exact: $\approx 2.1667$ b. Trapezoidal Rule: $\approx 1.1719$ c. Simpson's Rule: $\approx 1.1667$
E)a. Exact: $\approx 1.1667$ b. Trapezoidal Rule: $\approx 1.1719$ c. Simpson's Rule: $\approx 1.1667$

Question 32

Use a table of integrals to find the indefinite integral $\int x ^ { 9 } e ^ { x ^ { 10 } } d x$ .

A) $\frac { 1 } { 9 } e ^ { x ^ { 10 } } + C$
B) $\frac { 1 } { 10 } e ^ { x ^ { 9 } } + C$
C) $\frac { 1 } { 10 } e ^ { x ^ { 10 } } + C$
D) $\frac { 1 } { 9 } e ^ { x ^ { 9 } } + C$
E) $\frac { 1 } { 10 } e ^ { x } + C$

Question 33

A body assimilates a 12-hour cold tablet at a rate modeled by $d C / d t = 8 - \ln \left( t ^ { 2 } - 2 t + 4 \right) , 0 \leq t \leq 12$ where $d C / d t$ is measured in milligrams per hour and $t$ is the time in hours. Use Simpson's Rule with $n = 8$ to estimate the total amount of the drug absorbed into the body during the 12 hours.

A) $\approx$ 58.915 mg
B) $\approx$ 68.915 mg
C) $\approx$ 38.915 mg
D) $\approx$ 48.915 mg
E) $\approx$ 78.915 mg

Question 34

The revenue (in dollars per year) for a new product is modeled by $R = 10,000 \left[ 1 - \frac { 1 } { \left( 1 + 0.1 ^ { 2 } \right) ^ { 1 / 2 } } \right]$ where t the time in years. Estimate the total revenue from sales of the product over its first 4 years on the market. Round your answer to nearest dollar

A)$10,821
B)$6579
C)$15,830
D)$1138
E)$3291

Question 35

Use a table of integrals to find the indefinite integral $\int \frac { \ln x } { x ( 8 + 5 \ln x ) } d x$ .

A) $\frac { 1 } { 25 } [ 5 \ln x - 8 \ln | 8 + 5 \ln x | ] + C$
B) $[ 8 \ln x - 5 \ln | 8 + 5 \ln x | ] + C$
C) $\frac { 1 } { 25 } [ 5 \ln x + 8 \ln | 8 + 5 \ln x | ] + C$
D) $\ln 13 x + C$
E) $\frac { 1 } { 13 } \ln x + C$

Question 36

Use the table of integrals to find the average value of the growth function $N = \frac { 330 } { 1 + e ^ { 5.7 - 0.25 t } }$ over the interval $[ 22,27 ]$ , where N the size of a population and t is the time in days. Round your answer to three decimal places.

A)200.507
B)758.790
C)198.507
D)391.543
E)321.407

Question 37

Use a table of integrals with forms involving $\sqrt { a ^ { 2 } - u ^ { 2 } }$ to find $\int \frac { - 3 } { x ^ { 2 } \sqrt { 49 - x ^ { 2 } } } d x$

A) $\frac { 3 \sqrt { 49 - x ^ { 2 } } } { 49 x } + C$
B) $- \frac { \sqrt { 49 - x ^ { 2 } } } { 49 x } + C$
C) $\frac { 3 } { 7 } \ln \left| \frac { 7 + \sqrt { 49 - x ^ { 2 } } } { x } \right| + C$
D) $- \frac { 3 \sqrt { 49 - x ^ { 2 } } } { 49 x } + C$
E) $- \frac { 3 } { 7 } \ln \left| \frac { 7 + \sqrt { 49 - x ^ { 2 } } } { x } \right| + C$

Question 38

Use a table of integrals to find the indefinite integral $\int \frac { x ^ { 2 } } { ( 8 + 2 x ) ^ { 7 } } d x$ .

A) $\left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } + \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } - \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$
B) $\frac { 1 } { 8 } \left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } + \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } - \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$
C) $\frac { 1 } { 8 } \left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } + \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } + \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$
D) $\frac { 1 } { 8 } \left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } - \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } + \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$
E) $\left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } - \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } + \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$

Question 39

A body assimilates a 12-hour cold tablet at a rate modeled by $\frac { d C } { d t } = 6 - \ln \left( t ^ { 2 } - 2 t + 4 \right)$ , $0 \leq t \leq 12$ where t is measured in milligrams per hour and t is the time in hours. Use Simpson's Rule with n = 12 to estimate the total amount of the drug absorbed into the body during the 12 hours.

A)46.88
B)64.90
C)34.88
D)36.90
E)50.90

Question 40

Approximate the integral using Simpson's Rule: $\int _ { 0 } ^ { 5 } \frac { x } { 2 + x + x ^ { 2 } } d x$ , n = 6. Round your answer to three decimal places.

A)0.652
B)0.850
C)1.161
D)1.017
E)1.284

Question 41

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $\int _ { - \infty } ^ { 0 } e ^ { 6 x } d x$

A)converges to 0
B)converges to 6
C)converges to $\frac { 1 } { 6 }$
D)diverges to $- \infty$
E)diverges to $\infty$

Question 42

Suppose the mean height of American women between the ages of 30 and 39 is 64.5 inches, and the standard deviation is 2.7 inches. Use a symbolic integration utility to approximate the probability that a 30-to 39-year-old woman chosen at random is between 5 feet 4 inches and 6 feet tall.

A)0.8547
B)0.5707
C)0.4257
D)0.5734
E)0.9522

Question 43

A business is expected to yield a continuous flow of profit at the rate of $600,000 per year. If money will earn interest at the nominal rate of 8% per year compounded continuously, what is the present value of the business forever?

A)$7,600,000
B)$7,510,000
C)$7,500,000
D)$750,000
E)$850,000

Question 44

Evaluate the improper integral if it converges, or state that it diverges. $\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 8 } } d x$

A) $\frac { 1 } { 9 }$
B) $9$
C) $8$
D) $\frac { 1 } { 7 }$
E)diverges

Question 45

Evaluate the improper integral if it converges, or state that it diverges. $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 8 ] { x } } d x$

A) $\sqrt [ 8 ] { 9 }$
B) $1$
C) $8$
D) $9$
E)diverges

Question 46

The capitalized cost $C$ of an asset is given by $C = C _ { 0 } + \int _ { 0 } ^ { n } C ( t ) e ^ { - r t } d t$ where $C _ { 0 }$ is the original investment, $t$ is the time in years, $r$ is the annual interest rate compounded continuously, and $C ( t )$ is the annual cost of maintenance (in dollars). Find the capitalized cost of an asset (a) for 5 years, (b) for 10 years, and (c) forever. $C _ { 0 } = \$ 300,000 , C ( t ) = 15,000 , r = 6 \%$

A)a. For $n = 5 , C$ $\approx$ $253,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $4,466,666.67
B)a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $1,466,666.67
C)a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $2807,922.43c. For $n = \infty , C$ $\approx$ $4,466,666.67
D)a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $4,466,666.67
E)a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $466,666.67

Question 47

Find the capitalized cost C of an asset forever. The capitalized cost is given by $C = C _ { 0 } + \int _ { 0 } ^ { n } c ( t ) e ^ { - r t } d t$ where $C _ { 0 } = \$ 500,000$ is the original investment, t is the time in years, r = 12% is the annual interest rate compounded continuously, n is the total time in years over which the asset is capitalized, and $c ( t ) = 25,000 ( 1 + 0.08 t )$ is the annual cost of maintenance (measured in dollars). Round your answer to two decimal places.

A)$1,125,000.00
B)$875,000.00
C)$708,333.33
D)$899,218.75
E)$847,222.22

Question 48

Decide whether the integral is proper or improper. $\int _ { 0 } ^ { 5 } e ^ { - x } d x$

A)The integral is improper.
B)The integral is proper.

Question 49

Determine the amount of money required to set up a charitable endowment that pays the amount $P$ each year indefinitely for the annual interest rate compounded continuously. $P = \$ 12,000 , r = 6 \%$

A)$210,000
B)$200,000
C)$220,000
D)$240,000
E)$230,000

Question 50

Decide whether the following integral is improper. $\int _ { 0 } ^ { 1 } \frac { 1 } { 3 x - 2 } d x$

A)no
B)yes