Question 1

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

Accepted Answer

A) True 
 B)False

Question 2

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

Accepted Answer

The answer of Evaluate \(\int \left( \frac { x }...

Question 3

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

Accepted Answer

A) True 
 B)False

Question 4

Find the volume of the solid of revolution formed by rotating the region R about the x-axis.R is the region under the curve $y = \sqrt { 49 - x ^ { 2 } }$ from x = -7 to x = 7.

Accepted Answer

A)  $343 \pi$ 
B)  $\frac { 1,372 } { 3 } \pi$ 
C)  $49 \pi$ 
D)  $\frac { 196 } { 3 } \pi$ 
A)  $343 \pi$ 
B)  $\frac { 1,372 } { 3 } \pi$ 
C)  $49 \pi$ 
D)  $\frac { 196 } { 3 } \pi$

Question 5

Money is transferred continuously into an account at the constant rate of $1,100 per year. The account earns interest at the annual rate of 6% compounded continuously. How much will be in the account at the end of 4 years?

Accepted Answer

The answer of Money is transferred continuously into an account...

Question 6

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

Accepted Answer

A)  $\frac { 1 } { 3 } e ^ { 9 x } + C$ 
B)  $\frac { 3 } { 8 } e ^ { 8 x } + C$ 
C)  $3 e ^ { 8 x } + C$ 
D)  $24 e ^ { 8 x } + C$ 
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 7

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

Accepted Answer

A) True 
 B)False

Question 8

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?

Accepted Answer

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

Question 9

The area of the region bounded by the curves $y = x ^ { 2 }$ and $y = x ^ { 4 }$ is $\frac { 4 } { 15 }$ .

Accepted Answer

A) True 
 B)False

Question 10

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.

Accepted Answer

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

Question 11

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.

Accepted Answer

A) True 
 B)False

Question 12

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

Accepted Answer

A) True 
 B)False

Question 13

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

Accepted Answer

The answer of Evaluate \(\int \sqrt { 3 x +...

Question 14

Evaluate $\int \left( 9 x ^ { 4 } - 9 x + 4 \right) d x$ .

Accepted Answer

The answer of Evaluate \(\int \left( 9 x ^ {...

Question 15

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

Accepted Answer

The answer of Evaluate \(\int x ^ { 6 }...

Question 16

An investment will generate income continuously at the constant rate of $2,300 per year for 5 years. If the prevailing annual interest rate remains fixed at 10% compounded continuously, what is the present value of the investment?

Accepted Answer

A)  $9,049.79 
B)  $904.98 
C)  $90.50 
D)  $11,191.41 
A)  $9,049.79 
B)  $904.98 
C)  $90.50 
D)  $11,191.41

Question 17

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

Accepted Answer

The answer of Evaluate \(\int x ^ { 2 }...

Question 18

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

Accepted Answer

The answer of Evaluate \(\int 3 x ^ { 2...

Question 19

Find the average value of $f ( x ) = \frac { x ^ { 2 } + 1 } { x }$ over the interval $1 \leq x \leq e ^ { 2 }$ .

Accepted Answer

The answer of Find the average value of \(f (...

Question 20

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 1 to the end of minute 2?

Accepted Answer

The answer of An object is moving so that its...

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

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

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

Find the volume of the solid of revolution formed by rotating the region R about the x-axis.R is the region under the curve $y = \sqrt { 49 - x ^ { 2 } }$ from x = -7 to x = 7.

Money is transferred continuously into an account at the constant rate of $1,100 per year. The account earns interest at the annual rate of 6% compounded continuously. How much will be in the account at the end of 4 years?

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

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

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?

The area of the region bounded by the curves $y = x ^ { 2 }$ and $y = x ^ { 4 }$ is $\frac { 4 } { 15 }$ .

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 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.

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

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

Evaluate $\int \left( 9 x ^ { 4 } - 9 x + 4 \right) d x$ .

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

An investment will generate income continuously at the constant rate of $2,300 per year for 5 years. If the prevailing annual interest rate remains fixed at 10% compounded continuously, what is the present value of the investment?

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

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

Find the average value of $f ( x ) = \frac { x ^ { 2 } + 1 } { x }$ over the interval $1 \leq x \leq e ^ { 2 }$ .

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 1 to the end of minute 2?

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Exam 5: Integration

∫x3−3x2+5xdx=x33−3x22+5ln⁡x+C\int \frac { x ^ { 3 } - 3 x ^ { 2 } + 5 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 3 x ^ { 2 } } { 2 } + 5 \ln x + C∫xx3−3x2+5​dx=3x3​−23x2​+5lnx+C

Evaluate ∫(x3−2)2/3dx\int \left( \frac { x } { 3 } - 2 \right) ^ { 2 / 3 } d x∫(3x​−2)2/3dx .

∫x(4x−5x)dx=8x5/25−10x+C\int \sqrt { x } \left( 4 x - \frac { 5 } { x } \right) d x = \frac { 8 x ^ { 5 / 2 } } { 5 } - 10 \sqrt { x } + C∫x​(4x−x5​)dx=58x5/2​−10x​+C

Find the volume of the solid of revolution formed by rotating the region R about the x-axis.R is the region under the curve y=49−x2y = \sqrt { 49 - x ^ { 2 } }y=49−x2​ from x = -7 to x = 7.

Money is transferred continuously into an account at the constant rate of $1,100 per year. The account earns interest at the annual rate of 6% compounded continuously. How much will be in the account at the end of 4 years?

Evaluate the following integral: ∫3e8xdx\int 3 e ^ { 8 x } d x∫3e8xdx

∫(x+6)−11dx=−(x−6)−1212+C\int ( x + 6 ) ^ { - 11 } d x = - \frac { ( x - 6 ) ^ { - 12 } } { 12 } + C∫(x+6)−11dx=−12(x−6)−12​+C

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

The area of the region bounded by the curves y=x2y = x ^ { 2 }y=x2 and y=x4y = x ^ { 4 }y=x4 is 415\frac { 4 } { 15 }154​ .

Water flows into a tank at the rate of 2t+5\sqrt { 2 t + 5 }2t+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 study indicates that x months from now the population of a town will be increasing at the rate of 3+5xx3 + 5 x \sqrt { x }3+5xx​ people per month. The population will increase by 513 people during the next 9 months.

∫−10(5x3−7x+x3)dx=23\int _ { - 1 } ^ { 0 } \left( 5 x ^ { 3 } - 7 x + \sqrt [ 3 ] { x } \right) d x = \frac { 2 } { 3 }∫−10​(5x3−7x+3x​)dx=32​

Evaluate ∫3x+5dx\int \sqrt { 3 x + 5 } d x∫3x+5​dx .

Evaluate ∫(9x4−9x+4)dx\int \left( 9 x ^ { 4 } - 9 x + 4 \right) d x∫(9x4−9x+4)dx .

Evaluate ∫x6ex7+9dx\int x ^ { 6 } e ^ { x ^ { 7 } + 9 } d x∫x6ex7+9dx .

An investment will generate income continuously at the constant rate of $2,300 per year for 5 years. If the prevailing annual interest rate remains fixed at 10% compounded continuously, what is the present value of the investment?

Evaluate ∫x2ex3dx\int x ^ { 2 } e ^ { x ^ { 3 } } d x∫x2ex3dx .

Evaluate ∫3x2ex3dx\int 3 x ^ { 2 } e ^ { x ^ { 3 } } d x∫3x2ex3dx .

Find the average value of f(x)=x2+1xf ( x ) = \frac { x ^ { 2 } + 1 } { x }f(x)=xx2+1​ over the interval 1≤x≤e21 \leq x \leq e ^ { 2 }1≤x≤e2 .

An object is moving so that its velocity after t minutes is v(t)=4+6t+9t2v ( t ) = 4 + 6 t + 9 t ^ { 2 }v(t)=4+6t+9t2 meters per minute. How far does the object travel from the end of minute 1 to the end of minute 2?

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

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

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

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

Find the volume of the solid of revolution formed by rotating the region R about the x-axis.R is the region under the curve $y = \sqrt { 49 - x ^ { 2 } }$ from x = -7 to x = 7.

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

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

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?

The area of the region bounded by the curves $y = x ^ { 2 }$ and $y = x ^ { 4 }$ is $\frac { 4 } { 15 }$ .

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 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.

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

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

Evaluate $\int \left( 9 x ^ { 4 } - 9 x + 4 \right) d x$ .

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

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

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

Find the average value of $f ( x ) = \frac { x ^ { 2 } + 1 } { x }$ over the interval $1 \leq x \leq e ^ { 2 }$ .

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 1 to the end of minute 2?