Deck 5: Integration

Question 1

Evaluate $\int \left( 5 x ^ { 3 } - 5 x + 6 \right) d x$ .

A) $\frac { 5 x ^ { 9 } } { 9 } - \frac { 5 x ^ { 2 } } { 2 } + 6 x + C$
B) $40 x ^ { 7 } - 5 + C$
C) 5x⁹ -5x² +6x + C
D) $\frac { 5 x ^ { 9 } } { 9 } - \frac { 5 x ^ { 2 } } { 2 } + C$

Answer 1

$\frac { 5 x ^ { 9 } } { 9 } - \frac { 5 x ^ { 2 } } { 2 } + 6 x + C$

Question 2

Evaluate $\int \left( x ^ { 6 } - 7 x ^ { 2 } + 3 \right) d x$ .

Answer 2

$\frac { x ^ { 7 } } { 7 } - \frac { 7 x ^ { 3 } } { 3 } + 3 x + C$

Question 3

Evaluate $\int \left( 5 x ^ { 3 } - 3 x + 4 \right) d x$ .

A) $\frac { 5 x ^ { 4 } } { 4 } - \frac { 3 x ^ { 2 } } { 2 } + 4 x + C$
B) $15 x ^ { 2 } - 3 + C$
C) $5 x ^ { 4 } - 3 x ^ { 2 } + 4 x + C$
D) $\frac { 5 x ^ { 4 } } { 4 } - \frac { 3 x ^ { 2 } } { 2 } + C$

Answer 3

$\frac { 5 x ^ { 4 } } { 4 } - \frac { 3 x ^ { 2 } } { 2 } + 4 x + C$

Question 4

Evaluate $\int \left( 1 - \frac { 3 } { x } + \frac { 5 } { x ^ { 3 } } \right) d x$ .

Question 5

Evaluate $\int \left( 5 x ^ { 2 / 3 } - \frac { 3 } { x ^ { 5 } } + \frac { 1 } { 3 x } \right) d x$ .

Question 6

Evaluate $\int \left( 1 - \frac { 9 } { x } + \frac { 5 } { x ^ { 7 } } \right) d x$ .

Question 7

$\int \left( \frac { 5 } { x ^ { 2 } } - \frac { 2 } { x ^ { 3 } } + \frac { 5 } { \sqrt { x } } \right) d x = - \frac { 5 } { x } + \frac { 1 } { x ^ { 2 } } + 5 \ln | \sqrt { x } | + C$

Question 8

$\int \left( \frac { 9 } { x ^ { 2 } } - \frac { 2 } { x ^ { 3 } } + \frac { 8 } { \sqrt { x } } \right) d x = - \frac { 9 } { x } + \frac { 1 } { x ^ { 2 } } + 8 \ln | \sqrt { x } | + C$

Question 9

Evaluate $\int \left( 5 x ^ { 2 } - e ^ { x } \right) d x$ .

Question 10

Evaluate $\int \left( e ^ { - 3 x } + \frac { 5 } { x } \right) d x$ .

Question 11

Evaluate $\int \frac { 4 x - 1 } { x ^ { 5 } } d x$ .

Question 12

$\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 5 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 5 \ln x + C$

Question 13

Evaluate $\int \frac { 2 x - 3 } { x ^ { 4 } } d x$ .

Question 14

$\int \frac { x ^ { 3 } - 7 x ^ { 2 } + 4 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 7 x ^ { 2 } } { 2 } + 4 \ln | x | + C$

Question 15

Evaluate: $\int x ^ { 3 } ( 2 x + 5 - 3 \sqrt { x } ) d x$

A) $\frac { x ^ { 4 } } { 4 } \left( x ^ { 2 } + 5 x - 2 | x | \sqrt { x } \right) + C$
B) $\frac { 2 x ^ { 5 } } { 5 } + 5 x ^ { 3 } - \frac { 6 } { 5 } x ^ { 2 } \sqrt { x } + C$
C) $8 x ^ { 3 } + 15 x ^ { 2 } - \frac { 21 } { 2 } x ^ { 2 } \sqrt { x } + C$
D) $\frac { 2 x ^ { 5 } } { 5 } + \frac { 5 x ^ { 4 } } { 4 } - \frac { 2 } { 3 } x ^ { 4 } \sqrt { x } + C$

Question 16

$\int \sqrt { x } \left( 4 x - \frac { 5 } { x } \right) d x = \frac { 8 x ^ { 5 / 2 } } { 5 } - 10 \sqrt { x } + C$

Question 17

Evaluate $\int \left( x ^ { 3 } - 3 x ^ { 2 } + 5 \right) d x$ .

Question 18

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

Question 19

Evaluate $\int \left( e ^ { 2 t } + e ^ { - 3 t } \right) d t$ .

Question 20

A study indicates that x months from now the population of a certain city will be increasing at the rate of $( 3 + 4 x ) x ^ { - 1 / 2 }$ people per month. By how much will the population increase over the next 9 months?

A) 70 people
B) 80 people
C) 90 people
D) 100 people

Question 21

A study indicates that x months from now the population of a certain city will be increasing at the rate of $( 8 + 15 x ) x ^ { - 1 / 2 }$ people per month. By how much will the population increase over the next 16 months?

A) 704 people
B) 714 people
C) 694 people
D) 640 people

Question 22

A study indicates that x months from now the population of a town will be increasing at the rate of $3 + 5 x \sqrt { x }$ people per month. The population will increase by 513 people during the next 9 months.

Question 23

An object is moving along a straight line in such a way that after t minutes, its speed is $v ( t ) = 4 t ^ { 2 } + 3 t + 1$ meters per minute. How far does the object travel during the 3rd minute?

A) $\frac { 203 } { 6 }$ meters
B) $\frac { 205 } { 6 }$ meters
C) $\frac { 209 } { 6 }$ meters
D) $\frac { 211 } { 6 }$ meters

Question 24

An object is moving so that its velocity after t minutes is $v ( t ) = 4 + 6 t + 9 t ^ { 2 }$ meters per minute. How far does the object travel from the end of minute 4 to the end of minute 5?

A) 214 meters
B) 690 meters
C) 210 meters
D) 156 meters

Question 25

Specify the substitution you would choose to evaluate the integrals. $$\int \sqrt { 4 - 2 t } d t$$

A) u = t
B) $u = 4 - 2 t$
C) u = 2t
D) $u = \sqrt { 4 - 2 t }$

Question 26

Specify the substitution you would choose to evaluate the integrals. $$\int ( t + 4 ) e ^ { t ^ { 2 } + 8 t + 1 } d t$$

A) $u = t + 4$
B) $U = e ^ { t ^ { 2 } + 8 t + 1 }$
C) $u = t ^ { 2 } + 8 t + 1$
D) $u = ( t + 4 ) e$

Question 27

Specify the substitution you would choose to evaluate the integrals. $$\int \frac { \ln x ^ { 3 } } { 2 x } d x$$

A) $u = x ^ { 3 }$
B) $u = \frac { x ^ { 3 } } { 2 x }$
C) $u = 2 x$
D) $u = \ln x ^ { 3 }$

Question 28

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

A) $\frac { 49 } { 9 } \left( \frac { x } { 7 } - 2 \right) ^ { 9 / 7 } + C$
B) $\frac { x ^ { 2 } } { 14 } - 2 x + C$
C) $\frac { 7 } { 9 } \left( \frac { x } { 7 } - 2 \right) ^ { 9 / 7 } + C$
D) $\left( \frac { x } { 7 } - 2 \right) ^ { 9 / 7 } + C$

Question 29

$\int ( x + 6 ) ^ { 10 } d x = \frac { ( x + 6 ) ^ { 11 } } { 11 } + C$

Question 30

$\int ( x - 7 ) ^ { 8 } d x = \frac { ( x - 7 ) ^ { 9 } } { 9 } + C$

Question 31

$\int ( x + 3 ) ^ { - 5 } d x = - \frac { ( x + 3 ) ^ { - 6 } } { 6 } + C$

Question 32

$\int ( x + 5 ) ^ { 11 } d x = \frac { ( x + 5 ) ^ { 12 } } { 12 } + C$

Question 33

$\int ( x - 6 ) ^ { 7 } d x = \frac { ( x - 6 ) ^ { 8 } } { 8 } + C$

Question 34

$\int ( x + 6 ) ^ { - 11 } d x = - \frac { ( x - 6 ) ^ { - 12 } } { 12 } + C$

Question 35

Evaluate $\int \left( e ^ { - 8 x } + \frac { 8 } { x } \right) d x$ .

Question 36

Evaluate $\int \left( e ^ { 5 t } + e ^ { - 2 t } \right) d t$ .

Question 37

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

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

Question 38

Evaluate the following integral: $\int 3 e ^ { 8 x } d x$

A) $\frac { 1 } { 3 } e ^ { 9 x } + C$
B) $\frac { 3 } { 8 } e ^ { 8 x } + C$
C) $3 e ^ { 8 x } + C$
D) $24 e ^ { 8 x } + C$

Question 39

Evaluate $\int e ^ { 9 x - 6 } d x$ .

A) $\frac { e ^ { 9 ^ { x - 6 } } } { 9 } + C$
B) $e ^ { 9 x - 6 } + C$
C) $( 9 x - 6 ) e ^ { 9 x - 6 } + C$
D) $( 9 x - 6 ) C e ^ { 9 x - 6 }$

Question 40

$\int e ^ { - 5 x } d x = \frac { e ^ { - 5 x } } { 5 } + C$

Question 41

$\int e ^ { - 9 x } d x = \frac { e ^ { - 9 x } } { 9 } + C$

Question 42

Evaluate $\int \sqrt { 3 x + 7 } d x$ .

Question 43

Evaluate $\int \sqrt { 3 x + 5 } d x$ .

Question 44

Evaluate $\int \sqrt [ 2 ] { 7 x } d x$ .

A) $\frac { 2 ( 7 x ) ^ { 3 / 2 } } { 21 } + C$
B) $\frac { 7 x } { 2 } + C$
C) $\frac { 7 } { 2 ( 7 x ) ^ { 1 / 2 } } + C$
D) $\frac { 3 ( 7 x ) ^ { 3 / 2 } } { 14 } + C$

Question 45

Evaluate $\int \frac { 3 } { x - 4 } d x$ .

Question 46

Evaluate $\int \left( \frac { 1 } { 3 } x - 9 \right) ^ { 2 / 3 } d x$ .

A) $\frac { x ^ { 2 } } { 6 } - 9 x + C$
B) $\frac { 3 } { 5 } \left( \frac { 1 } { 3 } x - 9 \right) ^ { 5 / 3 } + C$
C) $\frac { 9 } { 5 } \left( \frac { 1 } { 3 } x - 9 \right) ^ { 5 / 3 } + C$
D) $\left( \frac { 1 } { 3 } x - 9 \right) ^ { 5 / 3 } + C$

Question 47

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

Question 48

Evaluate $\int 5 x ^ { 4 } e ^ { x ^ { 5 } } d x$ .

A) $e ^ {{ x } ^{6}}+ C$
B) $e ^ { x ^ { 5 } } + C$
C) $x ^ { 5 } e ^ { x ^ { 5 } } + C$
D) $25 x ^ { 5 } e ^ { x ^ { 5 } } + C$

Question 49

Evaluate $\int 9 x ^ { 8 } e ^ { x ^ { 9 } } d x$ .

A) $e ^ { x ^ { 9 } } + C$
B) $e ^ { x ^ { 10 } } + C$
C) $x ^ { 9 } e ^ { x ^ { 9 } } + C$
D) $81 x ^ { 9 } e ^ { x ^ { 9 } } + C$

Question 50

Evaluate $\int x ^ { 6 } e ^ { x ^ { 7 } + 9 } d x$ .

Question 51

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

Question 52

Evaluate $\int x ^ { 6 } \sqrt { x ^ { 7 } + 8 } d x$ .

A) $\frac { 2 \left( x ^ { 7 } + 8 \right) ^ { 3 / 2 } } { 21 } + C$
B) $\left( x ^ { 7 } + 8 \right) ^ { 3 / 2 } + C$
C) $C \left( x ^ { 7 } + 8 \right) ^ { 3 / 2 }$
D) $x ^ { 7 } + 8 + C$

Question 53

$\int 2 x \sqrt { x ^ { 2 } - 2 } d x = \frac { 2 \left( x ^ { 2 } - 2 \right) ^ { 3 / 2 } } { 3 } + C$

Question 54

Evaluate $\int x \sqrt { x ^ { 2 } + 4 } d x$ .

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

Question 55

Evaluate $\int x \sqrt { x ^ { 2 } + 9 } d x$ .

A) $\frac { \left( x ^ { 2 } + 9 \right) ^ { 3 / 2 } } { 3 } + C$
B) $\left( x ^ { 2 } + 9 \right) ^ { 3 / 2 } + C$
C) $\frac { x ^ { 3 } } { 3 } + 9 x + C$
D) $\frac { 3 \left( x ^ { 2 } + 9 \right) ^ { 3 / 2 } } { 4 } + C$

Question 56

Evaluate $\int \frac { x d x } { \sqrt { x ^ { 2 } + 1 } }$ .

Question 57

$\int 2 x \sqrt { x ^ { 2 } - 2 } d x = \frac { 2 \left( x ^ { 2 } - 2 \right) ^ { 3 / 2 } } { 3 } + C$

Question 58

Evaluate $\int x ^ { 2 } \left( x ^ { 3 } + 9 \right) ^ { 1 / 2 } d x$ .

A) $\frac { 2 } { 9 } \left( x ^ { 3 } + 9 \right) ^ { 3 / 2 } + C$
B) $\left( x ^ { 3 } + 9 \right) ^ { 3 / 2 } + C$
C) $C \left( x ^ { 3 } + 9 \right) ^ { 3 / 2 }$
D) $x ^ { 3 } + 9 + C$

Question 59

Evaluate $\int x ^ { 7 } e ^ { x ^ { 2 } + 3 } d x$ .

Question 60

Evaluate $\int \frac { x ^ { 3 } } { x ^ { 4 } + 2 } d x$ .

Question 61

Evaluate $\int \frac { x ^ { 5 } } { x ^ { 6 } + 3 } d x$ .

Question 62

Evaluate the indicated integral: $$\int \frac { 3 x ^ { 4 } + 60 x ^ { 3 } + 21 } { x ^ { 5 } + 25 x ^ { 4 } + 35 x + 4 } d x$$

A) $\frac { 3 } { 5 } \ln \left| x ^ { 4 } + 20 x ^ { 3 } + 7 \right| + C$
B) $\frac { 3 } { 5 } \ln \left| x ^ { 5 } + 25 x ^ { 4 } + 28 x \right| + C$
C) $\frac { 5 } { 3 } \ln \left| x ^ { 5 } + 25 x ^ { 4 } + 35 x + 4 \right| + C$
D) $\frac { 3 } { 5 } \ln \left| x ^ { 5 } + 25 x ^ { 4 } + 35 x + 4 \right| + C$

Question 63

Evaluate $\int \frac { x } { \sqrt { x ^ { 2 } + 6 } } d x$ .

Question 64

Solve the given initial value problem for y = f (x). $$\frac { d y } { d x } = \frac { x - 5 } { x ^ { 2 } - 10 x + 8 }$$ where y = 6 when x = 2

A) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right|$
B) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| + 6$
C) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| - \frac { 1 } { 2 } \ln 8 + 6$
D) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| - \frac { 1 } { 2 } \ln 8$

Question 65

The slope $$f ^ { \prime } ( x )$$ at each point (x, y) on a curve y = f (x) is given, along with a particular point (a,b) on the curve. Use this information to find f (x).

A) $y = - \frac { 1 } { 2 } e ^ { 16 - x ^ { 2 } }$
B) $y = - \frac { 3 } { 2 } - \frac { 1 } { 2 } e ^ { 16 - x ^ { 2 } }$
C) $y = - 2 - \frac { 1 } { 2 } e ^ { 16 - x ^ { 2 } }$
D) $y = 2 - \frac { 1 } { 2 } e ^ { 16 - x ^ { 2 } }$

Question 66

The velocity $$v ( t ) ( \mathrm { t } ) = x ^ { \prime } ( t )$$ at time t of an object moving along the x axis is given, along with the initial position x(0) of the object. $$x ^ { \prime } ( t ) = - 2 ( 5 t + 1 ) ^ { 1 / 2 }$$ ; x(0) = 3
Find:
(a) The position x(t) at time t.
(b) The position of the object at time t = 6.
(c) The time when the object is at x = 2. Round answers for parts (b) and (c) to one decimal place.

A) (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 } + \frac { 49 } { 15 }$
(b) x(6) = -42.8
(c) t = 0.4

B) (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 }$
(b) x(6) = -46.0
(c) t = -0.2

C) (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 } + \frac { 49 } { 15 }$
(b) x(6) = 0.7
(c) t = -6.5

D) (a) $x ( t ) = - \frac { 4 } { 15 } ( 5 t + 1 ) ^ { 3 / 2 }$
(b) x(6) = 1.4
(c) t = -9.7

Question 67

Water flows into a tank at the rate of $\sqrt { 2 t + 5 }$ ft³/min. If the tank is empty when t = 0, how much water does it contain 6 minutes later? Express the answer to two decimal places.

A) 0.63 ft³
B) 19.64 ft³
C) 32.08 ft³
D) 24.74 ft³

Question 68

In a certain section of the country, the price of chicken is currently $2 per kilogram. It is estimated that x weeks from now, the price will be increasing at a rate of $2 \sqrt { x + 1 }$ cents per kilogram. How much will chicken cost 5 weeks from now?

A) $2.18 per kilogram
B) $0.19 per kilogram
C) $3.18 per kilogram
D) $1.19 per kilogram

Question 69

Evaluate $\int _ { - 1 } ^ { 1 } x ^ { 11 } d x$ .

A) 0
B) $\frac { 1 } { 11 }$
C) $\frac { 1 } { 22 }$
D) $- \frac { 1 } { 22 }$

Question 70

Evaluate $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } - 3 x ^ { 2 } + 5 \right) d x$ .

Question 71

Evaluate $\int _ { - 2 } ^ { 2 } \left( x ^ { 5 } + 3 x ^ { 3 } - 2 x \right) d x$ .

Question 72

Evaluate $\int _ { 0 } ^ { 1 } \left( 9 x ^ { 3 } - 6 x ^ { 2 } + 7 \right) d x$ .

Question 73

Evaluate $\int _ { - 5 } ^ { 5 } \left( 5 x ^ { 5 } + 2 x ^ { 3 } - 6 x \right) d x$ .

Question 74

Evaluate $\int _ { - 2 } ^ { 2 } \left( x ^ { 5 } + 3 x ^ { 3 } - 2 x \right) d x$ . Round answer to three decimal places, if necessary.

A) 50.123
B) 4
C) 0
D) 6.892

Question 75

$\int _ { - 1 } ^ { 0 } \left( 5 x ^ { 3 } - 7 x + \sqrt [ 3 ] { x } \right) d x = \frac { 2 } { 3 }$

Question 76

Evaluate $\int _ { 1 } ^ { 2 } \left( 1 - \frac { 3 } { x } + \frac { 5 } { x ^ { 3 } } \right) d x$ .

Question 77

Evaluate $\int _ { 1 } ^ { 2 } \left( 3 - \frac { 8 } { x } + \frac { 3 } { x ^ { 3 } } \right) d x$ .

Question 78

Evaluate $\int _ { 2 } ^ { 3 } \frac { 2 x - 3 } { x ^ { 4 } } d x$ .

Question 79

Evaluate $\int _ { 2 } ^ { 3 } \frac { 3 x - 2 } { x ^ { 5 } } d x$ . Approximate to four decimal places.

Question 80

Evaluate $\int _ { - 1 } ^ { 3 } ( 3 x - 5 ) ^ { 4 } d x$ . Express your answer as a decimal. Approximate to one decimal place, if necessary.

A) 2,250.2
B) 2,251.6
C) 2,252.8
D) 2,253.4