Question 1

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

Accepted Answer

The answer of Evaluate \(\int x ( x + 7...

Question 2

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.

Accepted Answer

(a) 2.0253

Question 3

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

Accepted Answer

A) True 
 B)False

Question 4

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.

Accepted Answer

A)  $509.79 
B)  $305.87 
C)  $1,019.58 
D)  $339.86 
A)  $509.79 
B)  $305.87 
C)  $1,019.58 
D)  $339.86

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

A 3-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.

Accepted Answer

The answer of A 3-hour movie runs continuously at a...

Question 8

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.

Accepted Answer

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

Question 9

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

Accepted Answer

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 $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 10 x + 1 ) + C$ 
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 10

$$f ( x ) = \left\{ \begin{array} { l l } 
2 e ^ { - 2 x } & \text { for } x \geq 0 \
0 & \text { for } x < 0
\end{array} \right.$$ is a probability density function.

Accepted Answer

A) True 
 B)False

Question 11

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.

Accepted Answer

The answer of The clothes dryers at a laundromat run...

Question 12

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 6 minutes to solve the puzzle.

Accepted Answer

The answer of Suppose the length of time, x, that...

Question 13

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

Accepted Answer

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)  $\frac { x ^ { 2 } } { 2 } + x ( x - 6 ) \ln ( 2 x ) + C$ 
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 14

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.

Accepted Answer

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

Question 15

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

Accepted Answer

A)  $\frac { 1 } { 60 }$ 
B)  $\frac { 1 } { 90 }$ 
C)  90 
D)  $\frac { 1 } { 92 }$ 
A)  $\frac { 1 } { 60 }$ 
B)  $\frac { 1 } { 90 }$ 
C)  90 
D)  $\frac { 1 } { 92 }$

Question 16

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

Accepted Answer

The answer of Evaluate \(\int x ^ { 7 }...

Question 17

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

Accepted Answer

A) True 
 B)False

Question 18

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 19 years? Round to the nearest hundredth.

Accepted Answer

A)  0.31 
B)  0.35 
C)  0.17 
D)  0.38 
A)  0.31 
B)  0.35 
C)  0.17 
D)  0.38

Question 19

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

Accepted Answer

The answer of Evaluate \(\int x ^ { 7 }...

Question 20

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 3 x } d x$ .

Accepted Answer

The answer of Evaluate \(\int _ { 1 } ^...

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

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.

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

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

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

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

$f ( x ) = \left\{ \begin{array} { l l } 2 e ^ { - 2 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ is a probability density function.

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.

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

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.

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

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

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

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

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 3 x } d x$ .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Exam 6: Additional Topics in Integration

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

Approximate the integral ∫12e1/xdx\int _ { 1 } ^ { 2 } e ^ { 1 / x } d x∫12​e1/xdx using (a) the trapezoidal rule and (b) Simpson's rule, both with 6 subintervals. Round your answer to five decimal places.

∫0∞x4ex5dx=15\int _ { 0 } ^ { \infty } \frac { x ^ { 4 } } { e ^ { x ^ { 5 } } } d x = \frac { 1 } { 5 }∫0∞​ex5x4​dx=51​

Evaluate ∫2∞1xln⁡xdx\int _ { 2 } ^ { \infty } \frac { 1 } { x \ln \sqrt { x } } d x∫2∞​xlnx​1​dx .

Evaluate ∫1∞e−7xdx\int _ { 1 } ^ { \infty } e ^ { - 7 x } d x∫1∞​e−7xdx .

Evaluate ∫x(x−2)9dx\int x ( x - 2 ) ^ { 9 } d x∫x(x−2)9dx .

f(x)={2e−2x for x≥00 for x<0f ( x ) = \left\{ \begin{array} { l l } 2 e ^ { - 2 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.f(x)={2e−2x0​ for x≥0 for x<0​ is a probability density function.

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.

Evaluate ∫(x−3)ln⁡(2x)dx\int ( x - 3 ) \ln ( 2 x ) d x∫(x−3)ln(2x)dx .

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.

Evaluate the improper integral: ∫3∞x2(x3+3)2dx\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } } { \left( x ^ { 3 } + 3 \right) ^ { 2 } } d x∫3∞​(x3+3)2x2​dx

Evaluate ∫x7ex4dx\int x ^ { 7 } e ^ { x ^ { 4 } } d x∫x7ex4dx .

f(x)={116x for 0≤x≤80 otherwise 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.f(x)={161​x​0​ for 0≤x≤8 otherwise ​ is a probability density function.

Evaluate ∫x7(x4−6)7dx\int x ^ { 7 } \left( x ^ { 4 } - 6 \right) ^ { 7 } d x∫x7(x4−6)7dx .

Evaluate ∫1∞e−3xdx\int _ { 1 } ^ { \infty } e ^ { - 3 x } d x∫1∞​e−3xdx .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

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

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.

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

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

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

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

$f ( x ) = \left\{ \begin{array} { l l } 2 e ^ { - 2 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ is a probability density function.

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

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

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

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

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

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 3 x } d x$ .