Deck 6: Additional Topics in Integration

Question 1

Evaluate $\int x e ^ { x / 3 } d x$ .

A) $\frac { 1 } { 3 } x e ^ { x / 3 } - e ^ { x / 3 } + C$
B) $\frac { 1 } { 3 } x e ^ { x / 3 } - \frac { 1 } { 9 } e ^ { x / 3 } + C$
C) $3 x e ^ { x / 3 } - 9 e ^ { x / 3 } + C$
D) $3 x e ^ { x / 3 } + 9 e ^ { x / 3 } + C$

Answer 1

$3 x e ^ { x / 3 } - 9 e ^ { x / 3 } + C$

Question 2

Evaluate $\int x e ^ { x / 8 } d x$ .

A) $8 x e ^ { x / 8 } - 64 e ^ { x / 8 } + C$
B) $8 x e ^ { x / 8 } + 64 e ^ { x / 8 } + C$
C) $\frac { x e ^ { x / 8 } } { 8 } + \frac { e ^ { x / 8 } } { 64 } + C$
D) $\frac { x e ^ { x / 8 } } { 8 } - \frac { e ^ { x / 8 } } { 64 } + C$

Answer 2

$8 x e ^ { x / 8 } - 64 e ^ { x / 8 } + C$

Question 3

Use integration by parts to evaluate the integral $\int ( 3 - x ) e ^ { x } d x$ .

A) $( 4 - x ) e ^ { x } + C$
B) $( 4 + x ) e ^ { x } + C$
C) $( 3 - x ) e ^ { x } + C$
D) $( - 4 - x ) e ^ { x } + C$

Answer 3

$( 4 - x ) e ^ { x } + C$

Question 4

Evaluate $\int ( 2 - 3 x ) e ^ { - 2 x } d x$ .

A) $e ^ { - 2 x } \left( \frac { 3 } { 2 } x - \frac { 1 } { 4 } \right) + C$
B) $x e ^ { - 2 x } + C$
C) $\frac { 3 } { 2 } x e ^ { - 2 x } + C$
D) $- 3 e ^ { - 2 x } + C$

Question 5

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

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

Question 6

$\int x \ln ( x + 2 ) d x = \left( \frac { x ^ { 2 } } { 2 } - 2 \right) \ln ( x + 2 ) - \frac { x ^ { 2 } } { 4 } + x + C$

Question 7

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

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

Question 8

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$ .

A) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 144 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$
B) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } - \frac { 1 } { 72 } \left( x ^ { 2 } - 3 \right) ^ { 9 } + C$
C) $\frac { x ^ { 2 } } { 16 } \left( x ^ { 2 } - 3 \right) ^ { 8 } + C$
D) $\frac { \left( x ^ { 2 } - 3 \right) ^ { 8 } x ^ { 3 } } { 24 } + C$

Question 9

Evaluate $\int x ( x + 7 ) ^ { 8 } d x$ .

Question 10

Evaluate $\int x ^ { 7 } \left( x ^ { 4 } - 6 \right) ^ { 7 } d x$ .

Question 11

Evaluate $\int x ( x + 2 ) ^ { 7 } d x$ .

Question 12

Evaluate $\int x ^ { 7 } \left( x ^ { 4 } - 10 \right) ^ { 7 } d x$ .

Question 13

Use integration by parts to evaluate the integral $\int \frac { x } { \sqrt { x + 8 } } d x$ .

A) $\frac { 2 } { 3 } ( x - 8 ) ^ { 1 / 2 } ( x + 16 ) + C$
B) $\frac { 2 } { 3 } ( x + 8 ) ^ { 1 / 2 } ( x - 16 ) + C$
C) $\frac { 3 } { 2 } ( x + 8 ) ^ { 1 / 2 } ( x - 16 ) + C$
D) $\frac { 2 } { 3 } ( x + 8 ) ^ { 3 / 2 } ( x - 16 ) + C$

Question 14

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

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

Question 15

Evaluate $\int x ^ { 2 } \ln x d x$ .

Question 16

Evaluate $\int \frac { \ln 3 x } { x ^ { 2 } } d x$ .

Question 17

Evaluate $\int \frac { \ln 9 x } { x ^ { 5 } } d x$ .

Question 18

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

Question 19

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

Question 20

Evaluate $\int x ^ { 3 } e ^ { x ^ { 2 } } d x$ .

Question 21

Evaluate $\int x ^ { 7 } e ^ { x ^ { 4 } } d x$ .

Question 22

Evaluate $\int \frac { x } { 2 x + 5 } d x$ .

Question 23

Evaluate $\int \frac { x } { 5 x + 2 } d x$ .

Question 24

Use the formula $\int ( \ln x ) ^ { n } d x = x ( \ln x ) ^ { n } - n \int ( \ln x ) ^ { n - 1 } d x$ to evaluate $\int ( \ln x ) ^ { 3 } d x$ .

Question 25

$\int ( \ln x ) ^ { 2 } d x = x ( \ln x ) ^ { 2 } - 2 x \ln x + 2 x + C$

Question 26

After t hours on the job, a factory worker can produce $130 t e ^ { - 0.5 t }$ units per hour. How many units does the worker produce during the first 4 hours? Round to two decimal places.

A) 211.12 units
B) 308.88 units
C) 731.12 units
D) 344.06 units

Question 27

If, after t hours on the job, a factory worker can produce $10 t e ^ { - 0.02 t }$ units per hour, then the worker produces 130 units during the first 5 hours.

Question 28

It is projected that t years from now the population of a city will be changing at the rate of $t ^ { 2 } e ^ { 0.01 t }$ thousand people per year. If the current population is 1 million, what will the population be 4 years from now?

A) 1,028,000
B) 1,000,022
C) 1,028,800
D) 1,021,984

Question 29

After t weeks, a charity is raising money at the rate of 5,000 t ln(t + 1) dollars per week. How much money is raised during the first 10 weeks? Round to the nearest ten dollars.

A) $493,000
B) $493,480
C) $493,520
D) $493,550

Question 30

Given an initial population, $P _ { 0 } = 14,000$ , a renewal rate, R = 100, and a survival function, $S ( t ) = t e ^ { - 0.2 t }$ , with time t measured in years, determine the population at the end of 11 years. Round to two decimal places.

A) 8,120.55
B) 824.60
C) 58,646.60
D) 18,677.26

Question 31

From time t = 0 to t = 5 an object's speed is given by the function $s ( t ) = t e ^ { 3 t }$ . Compute the distance travelled by the object during this time interval. Round your answer to two decimals.

A) 5,085,148.51
B) 5,085,134.38
C) 5,085,239.58
D) 5,085,138.25

Question 32

An object moving in a straight line has velocity $v ( t ) = t e ^ { - \sqrt { t } }$ meters per second. Is it true that in the first 4 seconds the object will have travelled $4 e ^ { 2 } + 12$ meters?

Question 33

After t seconds, an object is moving at the speed of $t e ^ { 1 - t }$ meters per second. If the object begins at 0 when t = 0, then the distance the object travels as a function of time is expressed $s ( t ) = - e ^ { 1 - t } ( t + 1 )$ .

Question 34

Approximate the integral $\int _ { 1 } ^ { 2 } e ^ { 1 / x } d x$ using (a) the trapezoidal rule and (b) Simpson's rule, both with 6 subintervals. Round your answer to five decimal places.

Question 35

Approximate the integral $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x$ using (a) the trapezoidal rule and (b) Simpson's rule, both with 4 subintervals. Round your answer to five decimal places.

Question 36

Determine how many subintervals are required to guarantee accuracy to within 0.00001 for the approximation of the integral $$\int _ { 1 } ^ { 2 } \ln \left( 1 + x ^ { 2 } \right) d x$$ using (a) the trapezoidal rule and (b) Simpson's rule.

Question 37

Assume a 6-year franchise is expected to generate profit at the rate of $\sqrt [ 3 ] { t ^ { 2 } + 13,000 }$ dollars per year. If, over the next 6 years, the prevailing annual interest rate remains fixed at 6%, 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) $509.79
B) $305.87
C) $1,019.58
D) $339.86

Question 38

Shortly after leaving on a road trip, two math majors realize that the car's odometer is broken. To estimate the distance they travel between 8 PM and 9 PM, they record speedometer readings every 10 minutes: $$\begin{array} { c | c | c | c | c | c | c | c | } \text { Time } & 8 : 00 & 8 : 10 & 8 : 20 & 8 : 30 & 8 : 40 & 8 : 50 & 9 : 00 \\\hline \text { Speed (mph) } & 55 & 69 & 47 & 69 & 39 & 78 & 50\end{array}$$
Using Simpson's rule and only the information in the table, get the best possible estimate of the distance they traveled between 8 PM and 9 PM. Round your answer to one decimal place.

Question 39

Evaluate $\int _ { 1 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x } }$ .

A) 0
B) 1
C) $\frac { 1 } { 3 }$
D) Diverges

Question 40

Evaluate $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 4 ] { x } } d x$

A) Diverges
B) 0
C) $\frac { 4 } { 3 }$
D) $\frac { 3 } { 4 }$

Question 41

Evaluate $$\int _ { 5} ^ { \infty } \frac { 1 } { x ^ { 2 } - x } d x$$ . $$\left( \text { Hint: } \frac { 1 } { x ^ { 2 } - x } = \frac { 1 } { x - 1 } - \frac { 1 } { x } \right)$$

Question 42

Evaluate $\int _ { 2 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x ( x - 1 ) } }$ .

Question 43

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 5 x } d x$ .

A) $\frac { 1 } { 5 e ^ { 5 } }$
B) $\frac { 1 } { 5 e }$
C) $e ^ { 5 }$
D) $5 e ^ { 5 }$

Question 44

Given that $\int _ { 0 } ^ { \infty } e ^ { - x ^ { 2 } } d x = \frac { \sqrt { \pi } } { 2 }$ , evaluate $\int _ { 0 } ^ { \infty } e ^ { - 9 x ^ { 2 } } d x$ .

Question 45

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 7 x } d x$ .

A) $\frac { 1 } { 7 e ^ { 7 } }$
B) $\frac { 1 } { 7 e }$
C) e⁷
D) $7 e ^ { 7 }$

Question 46

$\int _ { 0 } ^ { \infty } \frac { 1 } { ( 1 + x ) ^ { 3 } } d x = \frac { 1 } { 8 }$ .

Question 47

Evaluate the improper integral: $\int _ { 2 } ^ { \infty } \frac { x ^ { 2 } } { \left( x ^ { 3 } + 6 \right) ^ { 2 } } d x$

A) 42
B) $\frac { 1 } { 28 }$
C) $\frac { 1 } { 44 }$
D) $\frac { 1 } { 42 }$

Question 48

$\int _ { 0 } ^ { \infty } \frac { 2 x } { 1 + x ^ { 2 } } d x = 0$

Question 49

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

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

Question 50

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

A) 0
B) $\frac { 1 } { 2 \ln 2 }$
C) $- \frac { 1 } { 2 \ln 2 }$
D) Diverges

Question 51

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } \frac { 1 } { ( x - 7 ) \ln ( x - 7 ) } d x$ Round your answer to two decimal places, if necessary.

A) -inf
B) 0.00
C) 2.08
D) Undefined

Question 52

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 5 } / 6 } d x$ Round to two decimal places, if necessary.

A) 1.5
B) 2
C) 1
D) Undefined

Question 53

$\int _ { 0 } ^ { \infty } \frac { x ^ { 2 } } { e ^ { x ^ { 3 } } } d x = \frac { 1 } { 3 }$

Question 54

$\int _ { 0 } ^ { \infty } \frac { x ^ { 4 } } { e ^ { x ^ { 5 } } } d x = \frac { 1 } { 5 }$

Question 55

The long run capitalized cost of an asset that initially cost $C _ { 0 }$ dollars is given by $C = C _ { 0 } + \int _ { 0 } ^ { \infty } A ( t ) e ^ { - r t } d t$ where A(t) is the annual cost of maintenance and r is the annual rate of interest, compounded continuously. Find the long run capitalized cost, in dollars, in the case where $C _ { 0 } = \$ 5,000,000$ , A(t) = 5, 000(1 + 3t), and r = 0.08.

Question 56

A hospital patient receives 4 units of a certain drug per hour intravenously. The drug is eliminated exponentially, so that the fraction that remains in the patients body for t hours is $f ( t ) = e ^ { - t / 12 }$ . If treatment is continued indefinitely, approximately how many units of the drug will be in the patient's body in the long run?

A) 44 units
B) 48 units
C) 52 units
D) 56 units

Question 57

A hospital patient receives intravenously 6 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 $f ( t ) = e ^ { - t / 11 }$ . 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) 17 units
B) 60 units
C) 66 units
D) 0.55 units

Question 58

A certain nuclear power plant produces radioactive waste at the rate of 500 pounds per year. The waste decays exponentially at the rate of 1.5% per year. How many pounds of radioactive waste from the plant will be present in the long run? Round to two decimal places, if necessary.

Question 59

$f ( x ) = \left\{ \begin{array} { l l } 7 e ^ { - 7 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ is a probability density function.

Question 60

$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 16 } \sqrt { x } & \text { for } 0 \leq x \leq 8 \\0 & \text { otherwise }\end{array} \right.$ is a probability density function.

Question 61

Find k so that $f ( x ) = \left\{ \begin{array} { l } k x ^ { 4 } + 0.4 \text { for } 0 \leq x \leq 1 \\0 \quad \text { otherw ise }\end{array} \right.$ is a probability density function.

Question 62

$f ( x ) = \left\{ \begin{array} { l r } \frac { 1 } { 7 } & \text { for } 0 \leq x \leq 7 \\0 & \text { otherwise }\end{array} \right.$ is a probability density function for a particular random variable X. Use integration to find $P ( 0 \leq x \leq 3 )$ rounded to the nearest hundredth.

A) 0.15
B) 0.57
C) 0.43
D) 0.28

Question 63

$f ( x ) = \left\{ \begin{array} { l l } \frac { 5 } { x ^ { 6 } } & \text { if } x \geq 1 \\0 & \text { if } x < 1\end{array} \right.$ is a probability density function for a particular random variable X. Use integration to find $P ( X \geq 3 )$

Question 64

$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 2 } e ^ { - x / 2 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ is a probability density function for a particular random variable X. Use integration to find $P ( X \geq 4 )$

A) $e ^ { 2 }$
B) $1 + e ^ { 2 }$
C) $e ^ { - 2 } - 1$
D) $e ^ { - 2 }$

Question 65

The life span of car stereos manufactured by a certain company is measured by a random variable X that is exponentially distributed with a probability density function $$f ( x ) = \left\{ \begin{array} { l l } 0.2 e ^ { - 0.2 x } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$$ where x is the life span in years of a randomly selected stereo. What is the probability that the life span of a randomly selected stereo is between 5 and 16 years? Round to the nearest hundredth.

A) 0.33
B) 0.36
C) 0.29
D) 0.16

Question 66

The useful life X of a particular kind of machine is a random variable with density function $$f ( x ) = \left\{ \begin{array} { l l } \frac { 4 } { 15 } + \frac { 2 } { 3 x ^ { 2 } } & \text { if } 2 \leq x \leq 5 \\0 & \text { otherwise }\end{array} \right.$$ where x is the number of years a randomly selected machine stays in use. $$P ( X \leq 3 ) = \frac { 28 } { 45 }$$

Question 67

The clothes dryers at a laundromat run for 45 minutes. You arrive at the laundromat and find that all of the dryers are being used. Use an appropriate uniform density function to find the probability that a dryer chosen at random will finish its cycle within 5 minutes.

Question 68

Let X be a random variable that measures the age of a randomly selected virus in a particular population. Suppose X is exponentially distributed with a probability density function $$f ( x ) = \left\{ \begin{array} { l l } k e ^ { - k x } & \text { for } x \geq 0 \\0 & \text { otherw ise }\end{array} \right.$$ where x is the age of a randomly selected virus and k is a positive constant. Experiments indicate that it is four times as likely for a virus to be less than 2 days old as it is for it to be more than 2 days old. Use this information to determine k.

Question 69

Suppose the length of time, x, that it takes a chimpanzee to solve a simple puzzle is measured by a random variable X that is exponentially distributed with a probability density function $$f ( x ) = \left\{ \begin{array} { l l } \frac { 2 } { 3 } e ^ { - 2 x / 3 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$$ where x is in minutes. Find the probability that a randomly chosen chimpanzee will take more than 12 minutes to solve the puzzle.

A) $e ^ { - 8 } + 1$
B) $e ^ { - 8 }$
C) $e ^ { 8 }$
D) $e ^ { 8 } - 1$

Question 70

Suppose the length of time that it takes a person to complete a hedgerow maze is measured by a random variable X that is exponentially distributed with a probability density function $$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 4 } x e ^ { - x / 2 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$$ where x is the number of minutes a randomly selected person takes to complete the maze. Find the probability that a randomly chosen person will take less than 4 minutes to complete the maze.

Question 71

A 2.5-hour movie runs continuously at a local theater. You leave for the theater without first checking the show times. Use an appropriate uniform density function to find the probability that you will arrive at the theater within 4 minutes of the start of the film (before or after). Round to the nearest hundredth.

A) 0.95
B) 0.04
C) 0.03
D) 0.05

Question 72

Evaluate $\int x e ^ { x / 6 } d x$ .

Question 73

Use integration by parts to evaluate the integral $\int ( 2 - x ) e ^ { x } d x$ .

Question 74

Evaluate $\int x ( x + 7 ) ^ { 8 } d x$ .

Question 75

Use integration by parts to evaluate the integral $\int \frac { x } { \sqrt { x + 11 } } d x$ .

Question 76

Evaluate $\int x ^ { 2 } \ln x d x$ .

Question 77

After t hours on the job, a factory worker can produce $130 t e ^ { - 0.5 t }$ units per hour. How many units does the worker produce during the first 4 hours? Round to two decimal places.

Question 78

Given an initial population, $P _ { 0 } = 17,000$ , a renewal rate, R = 150, and a survival function, $S ( t ) = t e ^ { - 0.2 t }$ , with time t measured in years, determine the population at the end of 11 years. Round to two decimal places.

Question 79

From time t = 0 to t = 2 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.

Question 80

Assume a 7-year franchise is expected to generate profit at the rate of $\sqrt [ 3 ] { t ^ { 2 } + 18,000 }$ dollars per year. If, over the next 7 years, the prevailing annual interest rate remains fixed at 8%, 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.