Question 1

Evaluate $\int x ^ { 2 } \ln x d x$ .

Accepted Answer

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

Question 2

$f ( x ) = \left\{ \begin{array} { l l } 
\frac { 5 } { x ^ { 6 } } & \text { if } x \geq 1 \
0 & \text { if } x < 1
\end{array} \right.$ is a probability density function for a particular random variable X. Use integration to find $P ( X \geq 3 )$

Accepted Answer

The answer of \(f ( x ) = \left\{ \begin{array}...

Question 3

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

Accepted Answer

The answer of Evaluate \(\int \frac { \ln 3 x...

Question 4

Evaluate the improper integral: $$\int _ { 10 } ^ { \infty } \frac { 1 } { ( x - 3 ) \ln ( x - 3 ) } d x$$ Round your answer to two decimal places, if necessary.

Accepted Answer

The answer of Evaluate the improper integral: \[\int _ {...

Question 5

Evaluate $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 6 ] { x } } d x$

Accepted Answer

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

Question 6

$f ( x ) = \left\{ \begin{array} { l l } 
7 e ^ { - 7 x } & \text { for } x \geq 0 \
0 & \text { for } x < 0
\end{array} \right.$ is a probability density function.

Accepted Answer

A) True 
 B)False

Question 7

Evaluate $\int \frac { \ln 9 x } { x ^ { 5 } } d x$ .

Accepted Answer

The answer of Evaluate \(\int \frac { \ln 9 x...

Question 8

$f ( x ) = \left\{ \begin{array} { l l } 
\frac { 1 } { 2 } e ^ { - x / 2 } & \text { if } x \geq 0 \
0 & \text { if } x < 0
\end{array} \right.$ is a probability density function for a particular random variable X. Use integration to find $P ( X \geq 4 )$

Accepted Answer

A)  $e ^ { 2 }$ 
B)  $1 + e ^ { 2 }$ 
C)  $e ^ { - 2 } - 1$ 
D)  $e ^ { - 2 }$ 
A)  $e ^ { 2 }$ 
B)  $1 + e ^ { 2 }$ 
C)  $e ^ { - 2 } - 1$ 
D)  $e ^ { - 2 }$

Question 9

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

Accepted Answer

A)  $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( 5 x + 1 ) + C$ 
B)  $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 10 x + 1 ) + C$ 
C)  $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 5 x + 2 ) + C$ 
D)  $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( x + 10 ) + C$ 
A)  $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( 5 x + 1 ) + C$ 
B)  $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 10 x + 1 ) + C$ 
C)  $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 5 x + 2 ) + C$ 
D)  $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( x + 10 ) + C$

Question 10

$\int _ { 0 } ^ { \infty } \frac { x ^ { 2 } } { e ^ { x ^ { 3 } } } d x = \frac { 1 } { 3 }$

Accepted Answer

A) True 
 B)False

Question 11

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

Accepted Answer

A)  $6 x - \frac { x ^ { 2 } } { 2 } + x ( x - 6 ) \ln ( 2 x ) + C$ 
B)  $\frac { ( x - 3 ) ^ { 2 } } { 2 } \ln ( 2 x ) + C$ 
C)  $\frac { x ^ { 2 } } { 2 } + x ( x - 6 ) \ln ( 2 x ) + C$ 
D)  $3 x - \frac { x ^ { 2 } } { 4 } + \frac { x ( x - 6 ) \ln ( 2 x ) } { 2 } + C$ 
A)  $6 x - \frac { x ^ { 2 } } { 2 } + x ( x - 6 ) \ln ( 2 x ) + C$ 
B)  $\frac { ( x - 3 ) ^ { 2 } } { 2 } \ln ( 2 x ) + C$ 
C)  $\frac { x ^ { 2 } } { 2 } + x ( x - 6 ) \ln ( 2 x ) + C$ 
D)  $3 x - \frac { x ^ { 2 } } { 4 } + \frac { x ( x - 6 ) \ln ( 2 x ) } { 2 } + C$

Question 12

is a probability density function for a particular random variable $$f ( x ) = \left\{ \begin{array} { l r } 
\frac { 1 } { 3 } & \text { for } 0 \leq x \leq 3 \
0 & \text { otherwise }
\end{array} \right.$$ variable X . Use integration to find $P ( 0 \leq x \leq 1 )$ rounded to the nearest hundredth.

Accepted Answer

A)  0.67 
B)  0.22 
C)  0.12 
D)  0.33 
A)  0.67 
B)  0.22 
C)  0.12 
D)  0.33

Question 13

Use integration by parts to evaluate the integral $\int ( 2 - x ) e ^ { x } d x$ .

Accepted Answer

The answer of Use integration by parts to evaluate the...

Question 14

$\int x \ln ( x + 2 ) d x = \left( \frac { x ^ { 2 } } { 2 } - 2 \right) \ln ( x + 2 ) - \frac { x ^ { 2 } } { 4 } + x + C$

Accepted Answer

A) True 
 B)False

Question 15

Evaluate $\int x ^ { 2 } \ln x d x$ .

Accepted Answer

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

Question 16

Approximate the integral $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x$ using (a) the trapezoidal rule and (b) Simpson's rule, both with 4 subintervals. Round your answer to five decimal places.

Accepted Answer

(a) 1.4906

Question 17

Shortly after leaving on a road trip, two math majors realize that the car's odometer is broken. To estimate the distance they travel between 8 PM and 9 PM, they record speedometer readings every 10 minutes: $$\begin{array} { c | c | c | c | c | c | c | c | } 
\text { Time } & 8 : 00 & 8 : 10 & 8 : 20 & 8 : 30 & 8 : 40 & 8 : 50 & 9 : 00 \
\hline \text { Speed (mph) } & 55 & 69 & 47 & 69 & 39 & 78 & 50
\end{array}$$ 
Using Simpson's rule and only the information in the table, get the best possible estimate of the distance they traveled between 8 PM and 9 PM. Round your answer to one decimal place.

Accepted Answer

The answer of Shortly after leaving on a road trip,...

Question 18

Find k so that $f ( x ) = \left\{ \begin{array} { l } 
k x ^ { 4 } + 0.4 \text { for } 0 \leq x \leq 1 \
0 \quad \text { otherw ise }
\end{array} \right.$ is a probability density function.

Accepted Answer

The answer of Find k so that \(f ( x...

Question 19

Evaluate $\int \frac { x } { 2 x + 5 } d x$ .

Accepted Answer

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

Question 20

$$f ( x ) = \left\{ \begin{array} { l r } 
\frac { 1 } { 8 } & \text { for } 0 \leq x \leq 8 \
0 & \text { otherwise }
\end{array} \right.$$ is a probability density function for a particular random variable X . Use integration to find $P ( 1 \leq x \leq 4 )$ rounded to the nearest hundredth

Accepted Answer

The answer of \[f ( x ) = \left\{ \begin{array}...

Evaluate $\int x ^ { 2 } \ln x d x$ .

$f ( x ) = \left\{ \begin{array} { l l } \frac { 5 } { x ^ { 6 } } & \text { if } x \geq 1 \\0 & \text { if } x < 1\end{array} \right.$ is a probability density function for a particular random variable X. Use integration to find $P ( X \geq 3 )$

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

Evaluate the improper integral: $\int _ { 10 } ^ { \infty } \frac { 1 } { ( x - 3 ) \ln ( x - 3 ) } d x$ Round your answer to two decimal places, if necessary.

Evaluate $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 6 ] { x } } d x$

$f ( x ) = \left\{ \begin{array} { l l } 7 e ^ { - 7 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ is a probability density function.

Evaluate $\int \frac { \ln 9 x } { x ^ { 5 } } d x$ .

$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 2 } e ^ { - x / 2 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ is a probability density function for a particular random variable X. Use integration to find $P ( X \geq 4 )$

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

$\int _ { 0 } ^ { \infty } \frac { x ^ { 2 } } { e ^ { x ^ { 3 } } } d x = \frac { 1 } { 3 }$

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

is a probability density function for a particular random variable $f ( x ) = \left\{ \begin{array} { l r } \frac { 1 } { 3 } & \text { for } 0 \leq x \leq 3 \\0 & \text { otherwise }\end{array} \right.$ variable X . Use integration to find $P ( 0 \leq x \leq 1 )$ rounded to the nearest hundredth.

Use integration by parts to evaluate the integral $\int ( 2 - x ) e ^ { x } d x$ .

$\int x \ln ( x + 2 ) d x = \left( \frac { x ^ { 2 } } { 2 } - 2 \right) \ln ( x + 2 ) - \frac { x ^ { 2 } } { 4 } + x + C$

Evaluate $\int x ^ { 2 } \ln x d x$ .

Approximate the integral $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x$ using (a) the trapezoidal rule and (b) Simpson's rule, both with 4 subintervals. Round your answer to five decimal places.

Find k so that $f ( x ) = \left\{ \begin{array} { l } k x ^ { 4 } + 0.4 \text { for } 0 \leq x \leq 1 \\0 \quad \text { otherw ise }\end{array} \right.$ is a probability density function.

Evaluate $\int \frac { x } { 2 x + 5 } d x$ .

$f ( x ) = \left\{ \begin{array} { l r } \frac { 1 } { 8 } & \text { for } 0 \leq x \leq 8 \\0 & \text { otherwise }\end{array} \right.$ is a probability density function for a particular random variable X . Use integration to find $P ( 1 \leq x \leq 4 )$ rounded to the nearest hundredth

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 ∫x2ln⁡xdx\int x ^ { 2 } \ln x d x∫x2lnxdx .

Evaluate ∫ln⁡3xx2dx\int \frac { \ln 3 x } { x ^ { 2 } } d x∫x2ln3x​dx .

Evaluate the improper integral: ∫10∞1(x−3)ln⁡(x−3)dx\int _ { 10 } ^ { \infty } \frac { 1 } { ( x - 3 ) \ln ( x - 3 ) } d x∫10∞​(x−3)ln(x−3)1​dx Round your answer to two decimal places, if necessary.

Evaluate ∫1∞1x6dx\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 6 ] { x } } d x∫1∞​6x​1​dx

f(x)={7e−7x for x≥00 for x<0f ( x ) = \left\{ \begin{array} { l l } 7 e ^ { - 7 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.f(x)={7e−7x0​ for x≥0 for x<0​ is a probability density function.

Evaluate ∫ln⁡9xx5dx\int \frac { \ln 9 x } { x ^ { 5 } } d x∫x5ln9x​dx .

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

∫0∞x2ex3dx=13\int _ { 0 } ^ { \infty } \frac { x ^ { 2 } } { e ^ { x ^ { 3 } } } d x = \frac { 1 } { 3 }∫0∞​ex3x2​dx=31​

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

Use integration by parts to evaluate the integral ∫(2−x)exdx\int ( 2 - x ) e ^ { x } d x∫(2−x)exdx .

∫xln⁡(x+2)dx=(x22−2)ln⁡(x+2)−x24+x+C\int x \ln ( x + 2 ) d x = \left( \frac { x ^ { 2 } } { 2 } - 2 \right) \ln ( x + 2 ) - \frac { x ^ { 2 } } { 4 } + x + C∫xln(x+2)dx=(2x2​−2)ln(x+2)−4x2​+x+C

Evaluate ∫x2ln⁡xdx\int x ^ { 2 } \ln x d x∫x2lnxdx .

Approximate the integral ∫01ex2dx\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x∫01​ex2dx using (a) the trapezoidal rule and (b) Simpson's rule, both with 4 subintervals. Round your answer to five decimal places.

Find k so that f(x)={kx4+0.4 for 0≤x≤10 otherw ise f ( x ) = \left\{ \begin{array} { l } k x ^ { 4 } + 0.4 \text { for } 0 \leq x \leq 1 \\0 \quad \text { otherw ise }\end{array} \right.f(x)={kx4+0.4 for 0≤x≤10 otherw ise ​ is a probability density function.

Evaluate ∫x2x+5dx\int \frac { x } { 2 x + 5 } d x∫2x+5x​dx .

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 $\int x ^ { 2 } \ln x d x$ .

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

Evaluate the improper integral: $\int _ { 10 } ^ { \infty } \frac { 1 } { ( x - 3 ) \ln ( x - 3 ) } d x$ Round your answer to two decimal places, if necessary.

Evaluate $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 6 ] { x } } d x$

$f ( x ) = \left\{ \begin{array} { l l } 7 e ^ { - 7 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ is a probability density function.

Evaluate $\int \frac { \ln 9 x } { x ^ { 5 } } d x$ .

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

$\int _ { 0 } ^ { \infty } \frac { x ^ { 2 } } { e ^ { x ^ { 3 } } } d x = \frac { 1 } { 3 }$

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

Use integration by parts to evaluate the integral $\int ( 2 - x ) e ^ { x } d x$ .

$\int x \ln ( x + 2 ) d x = \left( \frac { x ^ { 2 } } { 2 } - 2 \right) \ln ( x + 2 ) - \frac { x ^ { 2 } } { 4 } + x + C$

Evaluate $\int x ^ { 2 } \ln x d x$ .

Approximate the integral $\int _ { 0 } ^ { 1 } e ^ { x ^ { 2 } } d x$ using (a) the trapezoidal rule and (b) Simpson's rule, both with 4 subintervals. Round your answer to five decimal places.

Find k so that $f ( x ) = \left\{ \begin{array} { l } k x ^ { 4 } + 0.4 \text { for } 0 \leq x \leq 1 \\0 \quad \text { otherw ise }\end{array} \right.$ is a probability density function.

Evaluate $\int \frac { x } { 2 x + 5 } d x$ .