Question 1

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

Accepted Answer

A) True 
 B)False

Question 2

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

Accepted Answer

The answer of Evaluate \(\int x ^ { 8 }...

Question 3

A new health club has just opened. Statistics suggest that the fraction of clients who will still be attending the club t months after their initial visit is given by $f ( t ) = e ^ { - 0.08 t }$ . The club initially accepts 1,000 people for membership and plans to accept new members at the rate of 50 per month. Approximately 680 people will be in the club 2 years from now.

Accepted Answer

A) True 
 B)False

Question 4

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

Accepted Answer

The answer of Evaluate \(\int \frac { x ^ {...

Question 5

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). $$f ^ { \prime } ( x ) = x e ^ { 36 - x ^ { 2 } }$$ ; (-6, 1)

Accepted Answer

A)  $y = - 1 - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$ 
B)  $y = - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$ 
C)  $y = \frac { 3 } { 2 } - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$ 
D)  $y = 1 - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$ 
A)  $y = - 1 - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$ 
B)  $y = - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$ 
C)  $y = \frac { 3 } { 2 } - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$ 
D)  $y = 1 - \frac { 1 } { 2 } e ^ { 36 - x ^ { 2 } }$

Question 6

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

Accepted Answer

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

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

Accepted Answer

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)  $\frac { 3 ( 7 x ) ^ { 3 / 2 } } { 14 } + C$ 
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 8

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

Accepted Answer

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

Question 9

Find the volume of the solid of revolution formed by rotating the region R about the x-axis. R is the region under the line y = 5x + 2 from x = 0 to x = 2.

Accepted Answer

The answer of Find the volume of the solid of...

Question 10

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

Accepted Answer

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

Question 11

Evaluate the indicated integral: $\int \frac { 2 x ^ { 4 } + 40 x ^ { 3 } + 6 } { x ^ { 5 } + 25 x ^ { 4 } + 15 x + 3 } d x$

Accepted Answer

The answer of Evaluate the indicated integral: \(\int \frac {...

Question 12

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?

Accepted Answer

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

Question 13

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

Accepted Answer

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)  $\frac { e ^ { 3 x - 2 } } { 3 } + C$ 
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 14

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

Accepted Answer

A) True 
 B)False

Question 15

Determine the area of the region bounded by the curves $y = 1 + 4 x - x ^ { 2 }$ and $y = 1 + x ^ { 2 }$ .

Accepted Answer

The answer of Determine the area of the region bounded...

Question 16

The population density r miles from the center of a certain city is given by the function $D ( r ) = 2,000 e ^ { - r ^ { 2 } / 100 }$ people per square mile. How many people live within 10 miles of the city's center?

Accepted Answer

The answer of The population density r miles from the...

Question 17

Solve the given initial value problem for y = f (x). $\frac { d y } { d x } = \frac { x - 4 } { x ^ { 2 } - 8 x + 7 }$ where y = -2 when x = -3

Accepted Answer

The answer of Solve the given initial value problem for...

Question 18

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

Accepted Answer

A) True 
 B)False

Question 19

The average value of $f ( x ) = \frac { x ^ { 2 } + x + 1 } { x + 3 }$ over the interval 0 $\le$ x$\le$ 2 is $\frac { 7 \ln \left( \frac { 5 } { 3 } \right) } { 2 } - 1$ .

Accepted Answer

A) True 
 B)False

Question 20

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

Accepted Answer

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

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

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

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

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). $f ^ { \prime } ( x ) = x e ^ { 36 - x ^ { 2 } }$ ; (-6, 1)

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

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

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

Find the volume of the solid of revolution formed by rotating the region R about the x-axis. R is the region under the line y = 5x + 2 from x = 0 to x = 2.

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

Evaluate the indicated integral: $\int \frac { 2 x ^ { 4 } + 40 x ^ { 3 } + 6 } { x ^ { 5 } + 25 x ^ { 4 } + 15 x + 3 } d x$

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?

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

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

Determine the area of the region bounded by the curves $y = 1 + 4 x - x ^ { 2 }$ and $y = 1 + x ^ { 2 }$ .

The population density r miles from the center of a certain city is given by the function $D ( r ) = 2,000 e ^ { - r ^ { 2 } / 100 }$ people per square mile. How many people live within 10 miles of the city's center?

Solve the given initial value problem for y = f (x). $\frac { d y } { d x } = \frac { x - 4 } { x ^ { 2 } - 8 x + 7 }$ where y = -2 when x = -3

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

The average value of $f ( x ) = \frac { x ^ { 2 } + x + 1 } { x + 3 }$ over the interval 0 $\le$ x $\le$ 2 is $\frac { 7 \ln \left( \frac { 5 } { 3 } \right) } { 2 } - 1$ .

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

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

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Exam 5: Integration

∫e−5xdx=e−5x5+C\int e ^ { - 5 x } d x = \frac { e ^ { - 5 x } } { 5 } + C∫e−5xdx=5e−5x​+C

Evaluate ∫x8x9+9dx\int x ^ { 8 } \sqrt { x ^ { 9 } + 9 } d x∫x8x9+9​dx .

Evaluate ∫x3x4+2dx\int \frac { x ^ { 3 } } { x ^ { 4 } + 2 } d x∫x4+2x3​dx .

The slope f′(x)f ^ { \prime } ( x )f′(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). f′(x)=xe36−x2f ^ { \prime } ( x ) = x e ^ { 36 - x ^ { 2 } }f′(x)=xe36−x2 ; (-6, 1)

Evaluate the indicated integral: ∫3x4+60x3+21x5+25x4+35x+4dx\int \frac { 3 x ^ { 4 } + 60 x ^ { 3 } + 21 } { x ^ { 5 } + 25 x ^ { 4 } + 35 x + 4 } d x∫x5+25x4+35x+43x4+60x3+21​dx

Evaluate ∫7x2dx\int \sqrt [ 2 ] { 7 x } d x∫27x​dx .

Evaluate ∫3x+7dx\int \sqrt { 3 x + 7 } d x∫3x+7​dx .

Find the volume of the solid of revolution formed by rotating the region R about the x-axis. R is the region under the line y = 5x + 2 from x = 0 to x = 2.

Evaluate ∫−13(3x−5)4dx\int _ { - 1 } ^ { 3 } ( 3 x - 5 ) ^ { 4 } d x∫−13​(3x−5)4dx . Express your answer as a decimal. Approximate to one decimal place, if necessary.

Evaluate the indicated integral: ∫2x4+40x3+6x5+25x4+15x+3dx\int \frac { 2 x ^ { 4 } + 40 x ^ { 3 } + 6 } { x ^ { 5 } + 25 x ^ { 4 } + 15 x + 3 } d x∫x5+25x4+15x+32x4+40x3+6​dx

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

Evaluate ∫e3x−2dx\int e ^ { 3 x - 2 } d x∫e3x−2dx .

∫x3−7x2+4xdx=x33−7x22+4ln⁡∣x∣+C\int \frac { x ^ { 3 } - 7 x ^ { 2 } + 4 } { x } d x = \frac { x ^ { 3 } } { 3 } - \frac { 7 x ^ { 2 } } { 2 } + 4 \ln | x | + C∫xx3−7x2+4​dx=3x3​−27x2​+4ln∣x∣+C

Determine the area of the region bounded by the curves y=1+4x−x2y = 1 + 4 x - x ^ { 2 }y=1+4x−x2 and y=1+x2y = 1 + x ^ { 2 }y=1+x2 .

The population density r miles from the center of a certain city is given by the function D(r)=2,000e−r2/100D ( r ) = 2,000 e ^ { - r ^ { 2 } / 100 }D(r)=2,000e−r2/100 people per square mile. How many people live within 10 miles of the city's center?

Solve the given initial value problem for y = f (x). dydx=x−4x2−8x+7\frac { d y } { d x } = \frac { x - 4 } { x ^ { 2 } - 8 x + 7 }dxdy​=x2−8x+7x−4​ where y = -2 when x = -3

∫(x+5)11dx=(x+5)1212+C\int ( x + 5 ) ^ { 11 } d x = \frac { ( x + 5 ) ^ { 12 } } { 12 } + C∫(x+5)11dx=12(x+5)12​+C

The average value of f(x)=x2+x+1x+3f ( x ) = \frac { x ^ { 2 } + x + 1 } { x + 3 }f(x)=x+3x2+x+1​ over the interval 0 ≤\le≤ x ≤\le≤ 2 is 7ln⁡(53)2−1\frac { 7 \ln \left( \frac { 5 } { 3 } \right) } { 2 } - 127ln(35​)​−1 .

Specify the substitution you would choose to evaluate the integrals. ∫ln⁡x32xdx\int \frac { \ln x ^ { 3 } } { 2 x } d x∫2xlnx3​dx

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

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

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

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

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). $f ^ { \prime } ( x ) = x e ^ { 36 - x ^ { 2 } }$ ; (-6, 1)

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

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

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

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

Evaluate the indicated integral: $\int \frac { 2 x ^ { 4 } + 40 x ^ { 3 } + 6 } { x ^ { 5 } + 25 x ^ { 4 } + 15 x + 3 } d x$

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?

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

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

Determine the area of the region bounded by the curves $y = 1 + 4 x - x ^ { 2 }$ and $y = 1 + x ^ { 2 }$ .

The population density r miles from the center of a certain city is given by the function $D ( r ) = 2,000 e ^ { - r ^ { 2 } / 100 }$ people per square mile. How many people live within 10 miles of the city's center?

Solve the given initial value problem for y = f (x). $\frac { d y } { d x } = \frac { x - 4 } { x ^ { 2 } - 8 x + 7 }$ where y = -2 when x = -3

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

The average value of $f ( x ) = \frac { x ^ { 2 } + x + 1 } { x + 3 }$ over the interval 0 $\le$ x $\le$ 2 is $\frac { 7 \ln \left( \frac { 5 } { 3 } \right) } { 2 } - 1$ .

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