Deck 13: The Integral

Question 1

Calculate the Riemann Sum for the integral using n = 4.
$\int _ { 1 } ^ { 5 } x ^ { 2 } d x$

A)16
B) 24
C) 30
D) 54
E) 26

Answer 1

30

Question 2

Evaluate the integral. $\int _ { 3 } ^ { 4 } \frac { 24 x ^ { 2 } } { x ^ { 3 } - 16 } \mathrm {~d} x$

A) $12 ( \ln 48 + \ln 11 )$
B) $8 ( \ln 48 - \ln 11 )$
C) $16 ( \ln 48 + \ln 11 )$
D) $12 ( \ln 48 - \ln 11 )$
E) $24 ( \ln 11 - \ln 48 )$

Answer 2

$8 ( \ln 48 - \ln 11 )$

Question 3

Calculate the total area of the region bounded by the line $y = 15 x ^ { 2 } + 13$ , the x-axis, and the lines $x = 9$ and $x = 16$ .

A)24,216
B) 966
C) 16,926
D) 50,596
E) 16,744

Answer 3

16,926

Question 4

The oxygen consumption of a bird embryo increases from the time the egg is laid through the time the chick hatches. In a typical galliform bird, the oxygen consumption, in milliliters per hour, can be approximated by
$c ( t ) = - 0.00272 t ^ { 3 } + 0.138 t ^ { 2 } - 0.8923 t + 0.1499 ( 6 \leq t \leq 30 )$ where t is time (in days) since the egg was laid. (An egg will typically hatch at around $t = 28$ .) Find the total amount of oxygen consumed during the ninth and tenth days ( $t = 8$ to $t = 10$ ). Round your answer to the nearest whole number.

A)64
B) 18
C) 50
D) 1,138
E) 467

Question 5

Use the given graph to estimate the left Riemann sum for the given interval with the stated number of subdivisions. $[ 0,3 ] ; n = 3$

A)Left Sum = 1
B) Left Sum = 2
C) Left Sum = - 2
D) Left Sum = 0
E) Left Sum = - 1

Question 6

The way Professor Waner drives, he burns gas at the rate of $1 - e ^ { - 3 t }$ gallons each hour, t hours after a fill-up. Find the number of gallons of gas he burns in the first 13 hours after a fill-up. Please round the answer to the nearest gallon.

A)16 gallons
B) 11 gallons
C) 14 gallons
D) 17 gallons
E) 13 gallons

Question 7

Evaluate the integral. $\int _ { - 3 } ^ { 3 } \left( x ^ { 2 } + 98 \right) \mathrm { d } x$

A)642
B) 615
C) 18
D) 606
E) 588

Question 8

Evaluate the integral. $\int _ { 7 } ^ { 15 } \left( x ^ { 3 } - 43 x \right) \mathrm { d } x$

A)8,272
B) 88
C) 12,056
D) 20,328
E) 3,792

Question 9

Evaluate the integral. $\int _ { 8 } ^ { 12 } \left( 12 x - 69 x ^ { 0.2 } \right) \mathrm { d } x$

A) $12 \left( \frac { 8 ^ { 2 } } { 2 } - \frac { 12 ^ { 2 } } { 2 } \right) - 69 \left( \frac { 12 ^ { 0.2 } } { 0.2 } - \frac { 8 ^ { 0.2 } } { 0.2 } \right)$
B) $12 \left( \frac { 12 ^ { 3 } } { 3 } - \frac { 8 ^ { 3 } } { 3 } \right) - 69 \left( \frac { 8 ^ { 1.2 } } { 1.2 } - \frac { 12 ^ { 1.2 } } { 1.2 } \right)$
C) $12 \left( \frac { 12 ^ { 2 } } { 2 } - \frac { 8 ^ { 2 } } { 2 } \right) + 69 \left( \frac { 12 ^ { 1.2 } } { 1.2 } - \frac { 8 ^ { 1.2 } } { 1.2 } \right)$
D) $12 \left( \frac { 12 ^ { 2 } } { 2 } - \frac { 8 ^ { 2 } } { 2 } \right) - 69 \left( \frac { 12 ^ { 1.2 } } { 1.2 } - \frac { 8 ^ { 1.2 } } { 1.2 } \right)$
E) $12 \left( \frac { 12 ^ { 2 } } { 2 } - \frac { 8 ^ { 2 } } { 2 } \right) + 69 \left( \frac { 12 ^ { 1.2 } } { 1.2 } + \frac { 8 ^ { 1.2 } } { 1.2 } \right)$

Question 10

Evaluate the integral. $\int _ { - 1.2 } ^ { 2 } 4 e ^ { x - 3 } \mathrm {~d} x$

A) $\left( e ^ { - 1 } - e ^ { - 4.2 } \right)$
B) $4 \left( e ^ { - 1 } - e ^ { - 1.2 } \right)$
C) $4 \left( e ^ { - 1 } + e ^ { - 4.2 } \right)$
D) $4 \left( e ^ { 2 } - e ^ { - 4.2 } \right)$
E) $4 \left( e ^ { - 1 } - e ^ { - 4.2 } \right)$

Question 11

A car traveling down a road has a velocity of $v ( t ) = 55 - 3 e ^ { - \frac { t } { 16 } }$ mph at time ?t hours. Find the total distance it travels from time $t = 1$ hours to time $t = 5$ hours. (Round your answer to the nearest mile.)

A)216 miles
B) 210 miles
C) 218 miles
D) 224 miles
E) 206 miles

Question 12

Evaluate the integral. $\int _ { 6 } ^ { 10 } 42 e ^ { x } d x$

A) $42 \left( e ^ { 11 } - e ^ { 7 } \right)$
B) $42 \left( e ^ { 9 } + e ^ { 5 } \right)$
C) $\left( e ^ { 10 } - e ^ { 6 } \right)$
D) $42 e ^ { 4 }$
E) $42 \left( e ^ { 10 } - e ^ { 6 } \right)$

Question 13

Evaluate the integral. $\int _ { 4 } ^ { 9 } 3 \sqrt { x } \mathrm {~d} x$

A)38
B) 85.5
C) 10
D) 19
E) 2

Question 14

Calculate the left Riemann sum for the function over the given interval using the given values of n. $f ( x ) = 6 x - 2$ over $[ 0,2 ] , n = 4$

A)Left Sum = - 5
B) Left Sum = 5
C) Left Sum = 4
D) Left Sum = 7
E) Left Sum = - 4

Question 15

Evaluate the integral. $\int _ { 0 } ^ { 2 } 3 x \sqrt { 4 x + 4 } \mathrm {~d} x$

A) $0.5 \left( 2 \cdot 12 ^ { 3 } - 0.1 \cdot \left( 12 ^ { 5 } + 2 ^ { 5 } \right) \right)$
B) $0.5 \left( 2 \cdot 12 + 0.1 \cdot \left( 12 ^ { \frac { 2 } { 3 } } + 2 ^ { \frac { 2 } { 3 } } \right) \right)$
C) $0.5 \left( 2 \cdot 12 ^ { \frac { 3 } { 2 } } - 0.1 \cdot \left( 12 ^ { \frac { 5 } { 2 } } - 4 ^ { \frac { 5 } { 2 } } \right) \right)$
D) $0.5 \left( 2 \cdot 12 ^ { \frac { 1 } { 2 } } - 0.1 \cdot \left( 12 ^ { \frac { 3 } { 2 } } - 4 ^ { \frac { 3 } { 2 } } \right) \right)$
E) $0.5 \left( 2 \cdot 12 ^ { \frac { 3 } { 2 } } - 0.1 \cdot \left( 12 ^ { \frac { 1 } { 2 } } - 4 ^ { \frac { 1 } { 2 } } \right) \right)$

Question 16

Evaluate the integral. $\int _ { - 1 } ^ { 4 } 3 x e ^ { 2 x ^ { 2 } + 3 } \mathrm {~d} x$

A) $0.75 \left( e ^ { 32 } - e ^ { 2 } \right)$
B) $0.75 \left( e ^ { 35 } - e ^ { 5 } \right)$
C) $0.75 \left( e ^ { 35 } + e ^ { 5 } \right)$
D) $1.75 \left( e ^ { 35 } + e ^ { 5 } \right)$
E) $1.75 \left( e ^ { 35 } - e ^ { 5 } \right)$

Question 17

Evaluate the integral. $\int _ { 5 } ^ { 9 } \left( \frac { 46 } { x ^ { 2 } } + 101 x \right) \mathrm { d } x$

A) $\left( \frac { 46 } { 9 } - \frac { 46 } { 5 } \right) + 50.5 \left( 9 ^ { 2 } - 5 ^ { 2 } \right)$
B) $\left( \frac { 46 } { 9 } - \frac { 46 } { 5 } \right) + 101 \left( 9 ^ { 2 } - 5 ^ { 2 } \right)$
C) $\left( \frac { 46 } { 5 } - \frac { 46 } { 9 } \right) - 101 \left( 9 ^ { 2 } - 5 ^ { 2 } \right)$
D) $\left( \frac { 46 } { 5 } - \frac { 46 } { 9 } \right) + 50.5 \left( 9 ^ { 2 } - 5 ^ { 2 } \right)$
E) $\left( \frac { 46 } { 5 } - \frac { 46 } { 9 } \right) + 50.5 ( 9 - 5 )$

Question 18

Calculate the total area of the region bounded by the line $y = 17 x$ , the x-axis, and the lines $x = 3$ and $x = 5$ . Round your answer to the nearest whole number.

A)272
B) 833
C) 136
D) 34
E) 17

Question 19

Evaluate the integral. $\int _ { - 1 } ^ { 1 } 3 ^ { x - 3 } \mathrm {~d} x$

A) $\frac { 8 } { 81 \log _ { 2 } 3 }$
B) $\frac { 81 } { 8 \ln 3 }$
C) $\frac { 81 } { 8 }$
D) $\frac { 81 } { 8 \log _ { 2 } 3 }$
E) $\frac { 8 } { 81 \ln 3 }$

Question 20

Evaluate the integral. $\int _ { 0 } ^ { \sqrt { 2 } } 2 x \sqrt { x ^ { 2 } + 9 } \mathrm {~d} x$

A) $\frac { 2 } { 3 } \left( 11 ^ { \frac { 3 } { 2 } } - 27 \right)$
B) $- \frac { 2 \cdot 11 ^ { \frac { 2 } { 3 } } } { 3 }$
C) $\frac { 2 \cdot 11 ^ { \frac { 3 } { 2 } } } { 3 }$
D) $\frac { 2 } { 3 } \left( 27 - 11 ^ { \frac { 3 } { 2 } } \right)$
E) $\frac { 2 } { 3 } \left( 11 ^ { \frac { 2 } { 3 } } - 27 \right)$

Question 21

The rate of change $r ( t )$ of the total number, in thousand of articles, of research articles in the prominent journal Physics Review written by researchers in Europe is shown in the graph, where t is time in years $t = 0$ represents the start of 1983). Use both left and right Riemann sums with 8 subdivisions to estimate the total number of articles in Physics Review written by researchers in Europe in the 16-year period beginning 1983. (Estimate each value of $$r ( t )$$ to the nearest 0.5). Use the sums to obtain an estimate of $$\int _ { 0 } ^ { 16 } r ( t ) d t$$ .

A)9,000 articles
B) 50,500 articles
C) 46,000 articles
D) 4,500 articles
E) 55,000 articles

Question 22

Calculate the left Riemann sums for the function over the given interval, using the given values of n. (When rounding, round answers to four decimal places.)
$f ( x ) = \frac { x } { 1 + x ^ { 2 } }$ over $[ 0,2 ] , n = 5$

A)Left sum = 0.3548
B) Left sum = 1.7738
C) Left sum = 8.8688
D) Left sum = 0.7095
E) Left sum = 1.4190

Question 23

Decide on what substitution to use, and then evaluate the given integral using a substitution. $\int 23.1 \sqrt { - 7 x + 9 } \mathrm {~d} x$

A) $2.2 ( - 7 x + 9 ) ^ { \frac { 2 } { 3 } } + C$
B) $- 2.2 \sqrt { - 7 x + 9 } + C$
C) $- 2.2 ( - 7 x + 9 ) ^ { \frac { 3 } { 2 } } + C$
D) $\frac { 80.85 } { \sqrt { - 7 x + 9 } } + C$
E) $- \frac { 80.85 } { \sqrt { - 7 x + 9 } } + C$

Question 24

Calculate the Riemann Sum for the integral using n = 4.
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Question 25

The normal distribution curve which models, distributions of data in a wide range of applications, is given by the function
$p ( x ) = \frac { 1 } { \sqrt { 2 \pi } } e ^ { - ( x - \mu ) ^ { 2 } / 2 \delta ^ { 2 } }$ where $\pi = 3.14159265 \ldots$ and $\sigma$ and $\mu$ are constants called the standard deviation and the mean, respectively. Its graph is shown in the figure. ?
In a survey, consumers were asked to rate a new toothpaste on a scale of 1-10. The resulting data are modeled by a normal distribution with $\mu = 4.1$ and $\sigma = 1.2$ . The percentage of consumers who gave the toothpaste a score between a and b on the section is given by $\int _ { a } ^ { b } p ( x ) d x$
Use a Riemann sum with n = 10 to estimate the percentage of customers who rated the toothpaste 5 or higher. (Use the range 4.5 to 10.5.) Round your answer to the nearest whole number.

A)45%
B) 46%
C) 47%
D) 48%
E) 49%

Question 26

Evaluate the given integral using the substitution.
$\int e ^ { - 4 x } \mathrm {~d} x ; u = - 4 x$

A) $- e ^ { - 4 x } + C$
B) $- 4 e ^ { - 4 x } + C$
C) $- \frac { 1 } { 4 } e ^ { - 4 x } + C$
D) $- \frac { 1 } { 4 } e ^ { - 8 x } + C$
E) $- e ^ { - 8 x } + C$

Question 27

Calculate the Riemann Sum for the integral using n = 10.
$\int _ { 3 } ^ { 4 } 3 x e ^ { 1.7 x } d x$
Round the answer to four decimal places.

A)4,257.7370
B) 5,335.1538
C) 5,261.3439
D) 5,187.5341
E) 42,577.3704

Question 28

Evaluate the integral. $\int \left( 1 + 24.3 e ^ { 2.7 x - 4 } \right) \mathrm { d } x$

A) $x + 9 e ^ { 2.7 x - 3 } + C$
B) $x + \frac { 24.3 e ^ { 2.7 x - 4 } } { 2.7 x - 4 } + C$
C) $x + 9 e ^ { 2.7 x - 4 } + C$
D) $x + 24.3 e ^ { 2.7 x - 4 } + C$
E) $x + 9 e ^ { 2.7 x - 4 }$

Question 29

The graph of the derivative $f ^ { \prime } ( t )$ of $f ( t )$ is shown. Compute the total change of $f ( t )$ over the interval $[ 2,5 ]$ .

A) $f ( 5 ) - f ( 2 ) = 21$
B) $f ( 5 ) - f ( 2 ) = 7.5$
C) $f ( 5 ) - f ( 2 ) = 5.25$
D) $f ( 5 ) - f ( 2 ) = 2.25$
E) $f ( 5 ) - f ( 2 ) = 10.5$

Question 30

The marginal cost function for the manufacture of portable CD players is given by
$C ^ { \prime } = 50 - \frac { x } { 500 }$
where x is the number of CD players manufactured. Use a Riemann sum with n = 5 to estimate the cost of producing the first 5 CD players. Round your answer to the nearest cent.

A)$249.96
B) $249.95
C) $224.98
D) $249.97
E) $249.98

Question 31

Calculate the Riemann Sum for the integral using $n = 4$ . $\int _ { 0 } ^ { 10 } e ^ { - x ^ { 2 } } d x$

Round to the nearest tenth.

A) 1.1
B) 1
C) 1.3
D) 2.1
E) 2.5

Question 32

A model rocket has upward velocity $v ( t ) = 60 t ^ { 2 } \frac { \mathrm { ft } } { \mathrm { s } }$ , t seconds after launch. Use a Riemann sum with n = 10 to estimate how high the rocket is 4 seconds after launch.

A)1094.4 feet
B) 921.6 feet
C) 1478.4 feet
D) 1267.2 feet
E) 960 feet

Question 33

Evaluate the integral. $\int ( - 8 x + 4 ) ^ { - 2 } d x$

A) $\frac { \left( - 8 x ^ { 2 } + 4 \right) ^ { - 1 } } { 8 } + C$
B) $\frac { ( - 8 x + 4 ) ^ { - 1 } } { 8 } + C$
C) $\frac { ( - 8 x + 4 ) ^ { - 3 } } { 16 } + C$
D) $\frac { ( - 8 x + 4 ) ^ { - 3 } } { 8 } + C$
E) $\frac { ( - 8 x + 4 ) ^ { - 1 } } { 16 } + C$

Question 34

A race car has a velocity of $v ( t ) = 600 \left( 1 - e ^ { - 1.6 t } \right) \frac { \mathrm { ft } } { \mathrm { s } }$ , t seconds after starting. Use a Riemann sum with n = 10 to estimate how far the car travels in the first 2 seconds. (Round your answer to the nearest whole number.)

A)780 feet
B) 713 feet
C) 895 feet
D) 576 feet
E) 846 feet

Question 35

Evaluate the integral. $\int x \left( x ^ { 2 } + 2 \right) ^ { 2.5 } \mathrm {~d} x$

A) $\frac { \left( x ^ { 2 } + 2 \right) ^ { 2.5 } } { 7 } + C$
B) $\frac { \left( x ^ { 2 } + 2 \right) ^ { 2.5 } } { 2.5 } + C$
C) $\frac { \left( x ^ { 2 } + 2 \right) ^ { 3.5 } } { 7 } + C$
D) $\frac { \left( x ^ { 2 } + 2 \right) ^ { 2.5 } } { 5 } + C$
E) $\frac { x \left( x ^ { 2 } + 2 \right) ^ { 3.5 } } { 3.5 } + C$

Question 36

Calculate the Riemann Sum for the integral using n = 5. Round your answers to the two decimal places. $\int _ { - 5 } ^ { 10 } e ^ { - x } d x$

A)156.19
B) 468.57
C) 148.41
D) 7.78
E) 23.33

Question 37

Calculate the Riemann Sum for the integral using n = 5.
$\int _ { 0 } ^ { 1 } \frac { 3 } { 4 + x } d x$ ?
Give the answer correct to two decimal places.

A)0.33
B) 0.81
C) 0.68
D) 0.65
E) 0.53

Question 38

Evaluate the integral.
$\int ( 3 x + 5 ) ^ { 5 } \mathrm {~d} x$

A) $\frac { ( 3 x + 5 ) ^ { 6 } } { 18 }$
B) $\frac { ( 3 x + 5 ) ^ { 6 } } { 18 } + C$
C) $\frac { ( 3 x + 5 x ) ^ { 6 } } { 6 } + C$
D) $\frac { ( 3 x + 5 ) ^ { 6 } } { 3 } + C$
E) $\frac { ( 3 x + 5 ) ^ { 6 } } { 6 } + C$

Question 39

Evaluate the integral. $\int x \left( 5 x ^ { 2 } + 2 \right) ^ { 3 } \mathrm {~d} x$

A) $\frac { ( 5 x + 2 ) ^ { 2 } } { 5 } + C$
B) $\frac { x \left( 5 x ^ { 2 } + 2 \right) ^ { 4 } } { 40 } + C$
C) $\frac { \left( 5 x ^ { 2 } + 2 \right) ^ { 4 } } { 40 } + C$
D) $\frac { \left( 5 x ^ { 2 } + 2 \right) ^ { 4 } } { 20 } + C$
E) $\frac { \left( 5 x ^ { 2 } + 2 \right) ^ { 4 } } { 40 }$

Question 40

Calculate the left Riemann sums for the function over the given interval, using the given values of n. (When rounding, round answers to four decimal places.) $f ( x ) = e ^ { - x }$ over $[ - 5,5 ] , n = 5$

A)Left sum = 343.2693
B) Left sum = 171.6346
C) Left sum = 1,716.3464
D) Left sum = 46.4564
E) Left sum = 34.3269

Question 41

Evaluate the integral. ? $\int \left( - e ^ { x } + x ^ { - 9 } - \frac { 1 } { 2 } \right) d x$ ?

A) $- e ^ { x } - \frac { x ^ { - 8 } } { 8 } - \frac { x } { 2 } + C$
B) $- e ^ { x } + \frac { x ^ { - 8 } } { 8 } - \frac { x } { 2 } + C$
C) $- x - \frac { x ^ { - 8 } } { 8 } - \frac { 1 } { 2 } + C$
D) $e ^ { x } - \frac { x ^ { - 8 } } { 8 } - \frac { 1 } { 2 } + C$
E) $e ^ { x } + \frac { x ^ { - 8 } } { 8 } + C$

Question 42

Evaluate the integral.
$\int \frac { e ^ { - 0.05 x } } { 1 - e ^ { - 0.05 x } } \mathrm {~d} x$

A) $\frac { \ln \left| 1 - e ^ { - 0.05 x } \right| } { 20 } - x + C$
B) $- 20 \ln \left| 1 - e ^ { - 0.05 x } \right| + x + C$
C) $20 \ln \left| 1 - e ^ { - 0.05 x } \right| - x + C$
D) $0.05 \ln \left| 1 - e ^ { - 0.05 x } \right| + x + C$
E) $- 0.05 \ln \left| 1 - e ^ { - 0.05 x } \right| - x + C$

Question 43

Evaluate the integral. $\int \left( \frac { 1 } { v ^ { 9 } } + \frac { 3 } { v } \right) d v$

A) $- \frac { v ^ { - 8 } } { 8 } + 3 \ln ( v ) + C$
B) $\frac { v ^ { - 8 } } { 8 } - 3 \ln | v | + C$
C) $- \frac { v ^ { - 8 } } { 8 } + 3 \ln | v | + C$
D) $- \frac { v ^ { - 10 } } { 10 } - 3 \ln | v | + C$
E) $- \frac { v ^ { - 10 } } { 10 } + 3 \ln ( v ) + C$

Question 44

Find $f ( x )$ if $f ( 1 ) = \frac { 11 } { 2 }$ and the tangent line at $( x , f ( x ) )$ has the slope $11 x e ^ { x ^ { 2 } - 1 }$ .

A) $\frac { 11 } { 2 } e ^ { x ^ { 2 } - 1 } + \frac { 11 } { 2 }$
B) $\frac { 1 } { 2 } e ^ { x ^ { 2 } - 1 } + C$
C) $\frac { 1 } { 2 } e ^ { x ^ { 2 } - 1 }$
D) $\frac { 11 } { 2 } e ^ { x ^ { 2 } - 1 }$
E) $11 e ^ { x ^ { 2 } - 1 }$

Question 45

Find $f ( x )$ if $f ( 0 ) = 0$ and the tangent line at $( x , f ( x ) )$ has the slope $x \left( x ^ { 2 } + 2 \right) ^ { 3 }$ .

A) $\frac { x ^ { 2 } + 2 } { 8 } + C$
B) $\frac { x ^ { 2 } + 2 } { 8 }$
C) $\frac { \left( x ^ { 2 } + 2 \right) ^ { 4 } } { 4 } + 4$
D) $\frac { \left( x ^ { 2 } + 2 \right) ^ { 4 } } { 8 } - 2$
E) $\frac { \left( x ^ { 2 } + 2 \right) ^ { 4 } } { 4 } - 4$

Question 46

Evaluate the integral. $\int \left( \frac { 4 } { x ^ { 0.2 } } - \frac { 5 } { x ^ { 1.1 } } \right) d x$

A) $\frac { x ^ { - 1.2 } } { 0.2 } + \frac { 50 } { x ^ { 2.1 } } + C$
B) $\frac { x ^ { - 1.2 } } { 0.2 } + \frac { 50 } { x ^ { 2.1 } }$
C) $\frac { x ^ { 0.8 } } { 0.2 } - \frac { 50 } { x ^ { 0.1 } } + C$
D) $\frac { x ^ { 0.8 } } { 0.2 } + \frac { 50 } { x ^ { 0.1 } } + C$
E) $\frac { x ^ { 0.8 } } { 0.2 } + \frac { 50 } { x ^ { 0.1 } }$

Question 47

Evaluate the integral. $\int x ^ { - 9 } d x$

A) $- \frac { x ^ { - 10 } } { 8 } + C$
B) $- \frac { x ^ { - 10 } } { 10 } + C$
C) $- \frac { x ^ { - 8 } } { 8 } + C$
D) $\frac { x ^ { - 10 } } { 10 } + C$
E) $\frac { x ^ { - 8 } } { 8 } + C$

Question 48

Evaluate the integral.
$\int x ( x - 2 ) ^ { 3 } \mathrm {~d} x$

A) $x ( x - 2 ) ^ { 5 } + ( x - 2 ) ^ { 4 } + C$
B) $\frac { x ( x - 2 ) ^ { 4 } } { 4 } - \frac { ( x - 2 ) ^ { 5 } } { 20 } + C$
C) $\frac { ( x - 2 ) ^ { 5 } } { 5 } - \frac { ( x - 2 ) ^ { 4 } } { 2 } + C$
D) $\frac { ( x - 2 ) ^ { 5 } } { 5 } + \frac { ( x - 2 ) ^ { 4 } } { 2 } + C$
E) $\frac { ( x - 2 ) ^ { 4 } } { 4 } - \frac { ( x - 2 ) ^ { 5 } } { 5 } + C$

Question 49

Evaluate the integral. $\int 2 x \sqrt { 4 x ^ { 2 } - 4 } d x$

A) $\frac { \sqrt { \left( 4 x ^ { 2 } - 4 \right) } } { 2 } + C$
B) $\frac { \left( 4 x ^ { 2 } - 4 \right) \sqrt { 4 x ^ { 2 } - 4 } } { 4 } + C$
C) $\frac { \left( 4 x ^ { 2 } - 4 \right) \sqrt { 4 x ^ { 2 } - 4 } } { 6 } + C$
D) $\frac { \left( 4 x ^ { 2 } - 4 \right) \sqrt { 4 x ^ { 2 } - 4 } } { 2 } + C$
E) $\frac { \left( 4 x ^ { 2 } - 4 \right) \sqrt { 4 x ^ { 2 } - 4 } } { 12 } + C$

Question 50

The velocity of a particle moving in a straight line is given by $v = t \left( t ^ { 2 } + 4 \right) ^ { 4 } + 5 t$ .
Given that the distance $s = 0$ at $t = 0$ , find an expression for s in terms of t without any unknown constants.

A) $s ( t ) = \frac { \left( t ^ { 2 } + 4 \right) ^ { 5 } } { 10 } + \frac { 5 t ^ { 2 } } { 2 } - 102.4$
B) $s ( t ) = \frac { \left( t ^ { 2 } + 4 \right) ^ { 5 } } { 10 } + \frac { 5 t ^ { 2 } } { 2 }$
C) $s ( t ) = \frac { \left( t ^ { 2 } + 4 \right) ^ { 5 } } { 2 } + \frac { 5 t ^ { 2 } } { 2 } - 102.4$
D) $s ( t ) = \frac { \left( t ^ { 2 } + 4 \right) ^ { 5 } } { 10 } + \frac { 5 t ^ { 2 } } { 2 } + 102.4$
E) $s ( t ) = \frac { \left( t ^ { 2 } + 4 \right) ^ { 5 } } { 2 } + \frac { 5 t ^ { 2 } } { 2 } + 102.4$

Question 51

Evaluate the integral. $\int \frac { - 2 x - 7 } { \left( x ^ { 2 } + 7 x + 7 \right) ^ { 3 } } \mathrm {~d} x$

A) $- \frac { \left( x ^ { 2 } + 7 x + 7 \right) ^ { - 2 } } { 2 } + C$
B) $\frac { \left( x ^ { 2 } + 7 x + 7 \right) ^ { - 3 } } { 2 } + C$
C) $\left( x ^ { 2 } + 7 x + 7 \right) ^ { - 2 } + C$
D) $\frac { \left( x ^ { 2 } + 7 x + 7 \right) ^ { - 2 } } { 2 } + C$
E) $- \frac { 1 } { x ^ { 2 } + 7 x + 7 } + C$

Question 52

Evaluate the integral.
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Question 53

Evaluate the integral. $\int \left( \frac { x ^ { 2.6 } } { 5 } - 3.4 \right) d x$

A) $\frac { x ^ { 3.6 } } { 18 } + C$
B) $\frac { x ^ { 3.6 } } { 18 } + 3.4 x + C$
C) $\frac { x ^ { 1.6 } } { 18 } - 3.4 x + C$
D) $\frac { x ^ { 3.6 } } { 18 } - 3.4 x$
E) $\frac { x ^ { 3.6 } } { 18 } - 3.4 x + C$

Question 54

The marginal cost of producing the xth roll of film is given by $7 + \frac { 9 } { ( x + 2 ) ^ { 2 } }$ .
The total cost to produce one roll is $700. Find the total cost function $C ( x )$ .

A) $C ( x ) = 7 x + \frac { 9 } { x + 2 } + 696$
B) $C ( x ) = 7 x - \frac { 9 } { x + 2 } + C$
C) $C ( x ) = 7 x + \frac { 9 } { x + 2 }$
D) $C ( x ) = 7 x - \frac { 9 } { x + 2 } + 696$
E) $C ( x ) = 7 x - \frac { 9 } { x + 2 }$

Question 55

The velocity of a particle moving in a straight line is given by $v = t \left( t ^ { 2 } + 6 \right) ^ { 4 } + 3 t$ .
Find an expression for the position s after time t.

A) $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + 3 t ^ { 2 } + C$
B) $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 5 } + \frac { 3 t ^ { 2 } } { 2 } + C$
C) $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + \frac { 3 t ^ { 2 } } { 2 } + C$
D) $s ( t ) = \frac { t \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + \frac { 3 t ^ { 2 } } { 2 } + C$
E) $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + \frac { 3 t } { 2 } + C$

Question 56

Evaluate the integral. $\int x ^ { 7 } d x$

A) $7 x ^ { 8 } + C$
B) $\frac { x ^ { 8 } } { 8 }$
C) $7 x ^ { 49 }$
D) 1
E) $\frac { x ^ { 8 } } { 8 } + C$

Question 57

Evaluate the integral. $\int x e ^ { - 4 x ^ { 2 } + 1 } \mathrm {~d} x$

A) $- 8 e ^ { 4 x ^ { 2 } + 1 } + C$
B) $- \frac { e ^ { - 4 x ^ { 2 } + 1 } } { 8 } + C$
C) $- \frac { e ^ { 4 x ^ { 2 } + 1 } } { 4 } + C$
D) $- e ^ { 4 x ^ { 2 } + 1 } + C$
E) $- \frac { x e ^ { 4 x ^ { 2 } + 1 } } { 8 } + C$

Question 58

Evaluate the integral. $\int \left( ( 6 x - 1 ) e ^ { 6 x ^ { 2 } - 2 x } + 5 x e ^ { x ^ { 2 } } \right) \mathrm { d } x$

A) $\frac { e ^ { 6 x ^ { 2 } - 2 x } + 5 x e ^ { x ^ { 2 } } } { 2 x } + C$
B) $\frac { e ^ { 6 x ^ { 2 } - 2 x } + 5 e ^ { x ^ { 2 } } } { 2 } + C$
C) $e ^ { 6 x ^ { 2 } - 2 x } + 10 e ^ { x ^ { 2 } } + C$
D) $e ^ { 6 x ^ { 2 } - 2 x } + 5 e ^ { x ^ { 2 } } + C$
E) $\frac { e ^ { 6 x ^ { 2 } - 2 x } + 5 e ^ { x ^ { 2 } } } { x } + C$

Question 59

Evaluate the integral mentally. $\int ( - 5 ) d x$

A) $5 x + c$
B) $5 x$
C) 0
D) $- 5 x$
E) $- 5 x + c$

Question 60

Evaluate the integral. $\int \left( 6 x ^ { 23 } - x ^ { - 6 } + 2 \right) d x$

A) $\frac { x ^ { 23 } } { 4 } + \frac { x ^ { - 7 } } { 7 } + 2 x + C$
B) $\frac { x ^ { 24 } } { 4 } - \frac { x ^ { - 5 } } { 5 } + 2 x + C$
C) $\frac { x ^ { 24 } } { 4 } + \frac { x ^ { - 5 } } { 5 } + 2 x$
D) $\frac { x ^ { 24 } } { 4 } + \frac { x ^ { - 5 } } { 5 } + 2 x + C$
E) $\frac { x ^ { 24 } } { 4 } + \frac { x ^ { - 5 } } { 5 } + C$

Question 61

The marginal cost of producing the xth box of computer disks is $2 + \frac { x ^ { 2 } } { 120,000 }$ and the fixed cost is $120,000. Find the cost function $C ( x )$ .

A) $C ( x ) = 2 x + \frac { x ^ { 3 } } { 360,000 } - 120,000$
B) $C ( x ) = 2 x + \frac { x ^ { 3 } } { 360,000 } + 120,000$
C) $C ( x ) = - 2 x + \frac { x ^ { 3 } } { 360,000 } + 120,000$
D) $C ( x ) = \frac { x ^ { 3 } } { 360,000 } - 120,000$
E) $C ( x ) = \frac { x ^ { 3 } } { 360,000 } + 120,000$

Question 62

Find $f ( x )$ if $f ( 8 ) = 24$ and the tangent line at $( x , f ( x ) )$ has slope x. ?

A) $f ( x ) = \frac { x ^ { 2 } } { 2 } - 8$
B) $f ( x ) = \frac { x ^ { 2 } } { 2 } - C$
C) $f ( x ) = - \frac { x ^ { 2 } } { 2 } + 8$
D) $f ( x ) = \frac { x ^ { 2 } } { 2 } + C$
E) $f ( x ) = \frac { x ^ { 2 } } { 2 } + 8$

Question 63

Find $f ( x )$ if $f ( 4 ) = - 8$ and the tangent line at $( x , f ( x ) )$ has slope $7 e ^ { x } + 4$ . ?

A) $f ( x ) = 7 e ^ { x } + 4 x - 7 e ^ { 4 } + 24$
B) $f ( x ) = 7 e ^ { x } + 4 x + 7 e ^ { 4 } + 24$
C) $f ( x ) = 7 e ^ { x } + 4 x + 7 e ^ { 4 } - 24$
D) $f ( x ) = 7 e ^ { x } + 4 x - 7 e ^ { 4 } - 24$
E) $f ( x ) = 7 e ^ { x } - 4 x - 7 e ^ { 4 } + 24$

Question 64

Evaluate the integral. $\int \frac { x + 3 } { x ^ { 4 } } d x$

A) $- \frac { 1 } { 3 x ^ { 2 } } - \frac { 1 } { x ^ { 2 } } + C$
B) $- \frac { 1 } { 2 x ^ { 2 } } - \frac { 1 } { x ^ { 3 } } + C$
C) $- \frac { 1 } { 2 x ^ { 2 } } + \frac { 1 } { x ^ { 3 } } + C$
D) $\frac { 1 } { 2 x ^ { 2 } } + \frac { 1 } { x ^ { 3 } } + C$
E) $\frac { 1 } { 2 x ^ { 2 } } - \frac { 1 } { x ^ { 3 } } + C$

Question 65

Your name is Francesca Dragonetti (an assistant of Galileo Galilei) and, to impress your boss, you toss a weight upward at 32 feet per second from the top of the Leaning Tower of Pisa (height 185 ft). Neglecting air resistance, find the weight's velocity and the height of the weight above the ground as a functions of time in seconds. ?

A) $v ( t ) = 10 t$ , $s ( t ) = 185 + 32 t - 5 t ^ { 2 }$
B) $v ( t ) = 32 + 32 t$ , $s ( t ) = 185 + 32 t + 16 t ^ { 2 }$
C) $v ( t ) = 32 - 32 t$ , $s ( t ) = 32 t - 16 t ^ { 2 }$
D) $v ( t ) = 32 - 32 t$ , $s ( t ) = 185 + 32 t - 16 t ^ { 2 }$
E) $v ( t ) = 32 t$ , $s ( t ) = 32 t - 16 t ^ { 2 }$

Question 66

The velocity of a particle moving in a straight line is given by $v = e ^ { 3 t } + 3 t$ .
Find an expression for the position s after a time t.

A) $s ( t ) = 3 e ^ { 3 t } + 3 + C$
B) $s ( t ) = \frac { 1 } { 3 } e ^ { 3 t } + \frac { 3 } { 2 } t ^ { 2 } + C$
C) $s ( t ) = \frac { 1 } { 3 } e ^ { 3 t } + 6 t ^ { 3 } + C$
D) $s ( t ) = \frac { 1 } { 3 } e ^ { 4 t } + \frac { 3 } { 2 } t ^ { 2 } + C$
E) $s ( t ) = e ^ { 3 t } + 3 t ^ { 2 } + C$