Deck 6: Additional Topics in Integration

Question 1

Evaluate the improper integral: $\int _ { 7 } ^ { \infty } \frac { 1 } { ( x - 5 ) \ln ( x - 5 ) } d x$ Round your answer to two decimal places,if necessary.

A)0.69
B)1.95
C)Undefined
D)-0.37

Answer 1

Undefined

Question 2

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 6 x } d x$ .

A)e⁶
B) $6 e ^ { 6 }$
C) $\frac { 1 } { 6 e }$
D) $\frac { 1 } { 6 e ^ { 6 } }$

Answer 2

$\frac { 1 } { 6 e ^ { 6 } }$

Question 3

Evaluate $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt { x } } d x$

A)0
B) $\frac { 7 } { 6 }$
C) $\frac { 6 } { 7 }$
D)Diverges

Answer 3

Diverges

Question 4

Evaluate $\int x e ^ { x y } d x$ .

A) $\frac { x e ^ { x / 9 } } { 9 } + \frac { e ^ { x / 9 } } { 81 } + C$
B) $9 x e ^ { x / 9 } - 81 e ^ { x / 9 } + C$
C) $\frac { x e ^ { x / 9 } } { 9 } - \frac { e ^ { x / 9 } } { 81 } + C$
D) $9 x e ^ { x / 9 } + 81 e ^ { x / 9 } + C$

Question 5

Evaluate $\int x ( x - 2 ) ^ { 9 } d x$ .

A) $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 5 x + 2 ) + C$
B) $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( 5 x + 1 ) + C$
C) $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 10 x + 1 ) + C$
D) $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( x + 10 ) + C$

Question 6

Evaluate $\int ( x - 3 ) \ln ( 2 x ) d x$ .

A) $6 x - \frac { x ^ { 2 } } { 2 } + x ( x - 6 ) \ln ( 2 x )$ + C
B) $\frac { ( x - 3 ) ^ { 2 } } { 2 } \ln ( 2 x )$ + C
C) $\frac { x ^ { 2 } } { 2 } + x ( x - 6 ) \ln ( 2 x )$ + C
D) $3 x - \frac { x ^ { 2 } } { 4 } + \frac { x ( x - 6 ) \ln ( 2 x ) } { 2 }$ + C

Question 7

Evaluate the improper integral: $\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } } { \left( x ^ { 3 } + 7 \right) ^ { 2 } } d x$

A) $\frac { 1 } { 104 }$
B) $\frac { 1 } { 102 }$
C) $\frac { 1 } { 68 }$
D)102

Question 8

From time t = 0 to t = 4 an object's speed is given by the function $s ( t ) = t e ^ { 4 t }$ .Compute the distance travelled by the object during this time interval.Round your answer to two decimals.

A)8,330,724.81
B)8,330,728.68
C)8,330,738.94
D)8,330,830.01

Question 9

Evaluate $\int x \ln \sqrt { x } d x$ .

A) $\frac { x ^ { 2 } \ln x } { 4 } + C$
B) $x ^ { 2 } \ln x + C$
C) $\frac { x ^ { 2 } \ln x } { 4 } - \frac { x ^ { 2 } } { 8 }$ + C
D)x + C

Question 10

Given an initial population, $P _ { 0 } = 11,000$ ,a renewal rate,R = 100,and a survival function, $S ( t ) = t e ^ { - 0.3 t }$ ,with time t measured in years,determine the population at the end of 13 years.Round to two decimal places.

A)171.99
B)12,231.87
C)3,895.50
D)1,693.69

Question 11

Find the average value of the function $f ( x ) = x e ^ { - x / 3 }$ over the interval from 0 to 4.Round to two decimal places.

A)0.87
B)3.46
C)10.86
D)7.79

Question 12

A hospital patient receives intravenously 15 units of a certain drug per hour.The drug is eliminated exponentially,so that the fraction that remains in the patients body for t hours is $f ( t ) = e ^ { - t / 6 }$ .If the treatment is continued indefinitely,approximately how many units of the drug will be in the patient's body in the long run? Round to two decimal places,if necessary.

A)90 units
B)2.50 units
C)75 units
D)21 units

Question 13

A manufacturer has found that marginal cost is $( 2 q + 1 ) e ^ { - 0.02 q }$ dollars per unit when q units have been produced.The total cost of producing the first 10 units is $100.What is the total cost of producing the first 30 units?

A)$545.39
B)$635.39
C)$1,124.68
D)$300.00

Question 14

Use integration by parts to evaluate the integral $\int \frac { x } { \sqrt { x + 16 } } d x$ .

A) $\frac { 2 } { 3 } ( x + 16 ) ^ { 3 / 2 } ( x - 32 ) + C$
B) $\frac { 2 } { 3 } ( x - 16 ) ^ { 1 / 2 } ( x + 32 ) + C$
C) $\frac { 2 } { 3 } ( x + 16 ) ^ { 1 / 2 } ( x - 32 ) + C$
D) $\frac { 3 } { 2 } ( x + 16 ) ^ { 1 / 2 } ( x - 32 ) + C$

Question 15

Evaluate $\int _ { 2 } ^ { \infty } \frac { 1 } { x ( \ln x ) ^ { 3 } } d x$ .

A) $- \frac { 1 } { 2 ( \ln 2 ) ^ { 2 } }$
B) $- \frac { 1 } { 2 \ln 2 }$
C) $\frac { 1 } { 2 \ln 2 }$
D) $\frac { 1 } { 2 ( \ln 2 ) ^ { 2 } }$

Question 16

After t hours on the job,a factory worker can produce $150 t e ^ { - 0.5 t }$ units per hour.How many units does the worker produce during the first 4 hours? Round to two decimal places.

A)843.60 units
B)397.00 units
C)356.40 units
D)243.60 units

Question 17

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } x ^ { 3 } e ^ { - x ^ { 4 } n } d x$ Round to two decimal places,if necessary.

A)1.75
B)Undefined
C)3.5
D)2.63

Question 18

Demographic studies conducted in a certain city indicate that the fraction of the residents that will remain in the city for at least t years is $f ( t ) = e ^ { - t / 50 }$ .The current population of the city is 500,000,and it is estimated that new residents will be arriving at the rate of 1,000 people per year.If this estimate is correct,what will be the population in the long run?

A)40,800
B)50,000
C)550,000
D)40,500

Question 19

Assume a 6-year franchise is expected to generate profit at the rate of $\sqrt [ 3 ] { t ^ { 2 } + 19,000 }$ dollars per year.If,over the next 6 years,the prevailing annual interest rate remains fixed at 7%,compounded continuously,what is the present value of the franchise? Use Simpson's rule with n = 6 to approximate the integral.Round your answer to two decimal places.

A)$1,123.47
B)$561.73
C)$374.49
D)$337.04

Question 20

Use integration by parts to evaluate the integral $\int ( 15 - x ) e ^ { x } d x$ .

A) $( 16 + x ) e ^ { x } + C$
B) $( - 16 - x ) e ^ { x } + C$
C) $( 16 - x ) e ^ { x } + C$
D) $( 15 - x ) e ^ { x } + C$