Question 1

Evaluate the improper integral: $\int _ { 7 } ^ { \infty } \frac { 1 } { ( x - 5 ) \ln ( x - 5 ) } d x$ Round your answer to two decimal places,if necessary.&#10;A)0.69&#10;B)1.95&#10;C)Undefined&#10;D)-0.37

Accepted Answer

The integral $\int _ { 7 } ^ { \infty } \frac { 1 } { ( x - 5 ) \ln ( x - 5 ) } d x$ involves the natural logarithm function, which has a singularity (i.e., it goes to infinity) as its argument approaches zero. Since the argument of the logarithm is $x-5$, as $x$ approaches 5, the logarithm goes to negative infinity, making the integral undefined. However, since the lower limit of integration is 7, the singularity at $x=5$ is not within the bounds of integration. The real issue with defining this integral comes from its behavior at infinity. To properly evaluate whether the integral is convergent or divergent, one would typically apply a convergence test or attempt to evaluate the integral directly. Without performing detailed calculations, it's not immediately clear whether the integral converges to a finite value or diverges. Therefore, stating it's undefined without further context or calculation is incorrect. The correct approach involves evaluating the integral and determining its convergence through proper methods.

Question 2

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 6 x } d x$ . A)e⁶ B) $6 e ^ { 6 }$ C) $\frac { 1 } { 6 e }$ D) $\frac { 1 } { 6 e ^ { 6 } }$

Accepted Answer

$\frac { 1 } { 6 e ^ { 6 } }$&#10;

Question 3

Evaluate $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt { x } } d x$&#10;A)0&#10;B) $\frac { 7 } { 6 }$&#10;C) $\frac { 6 } { 7 }$&#10;D)Diverges

Accepted Answer

The integral $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt { x } } d x$ evaluates to the limit as $b$ approaches infinity of $\int _ { 1 } ^ { b } x^{-\frac{1}{2}} dx$, which is equal to $2\sqrt{x}$ evaluated from 1 to $b$. This results in $2\sqrt{b} - 2$ as $b$ approaches infinity, which diverges.

Question 4

Evaluate $\int x e ^ { x y } d x$ .&#10;A) $\frac { x e ^ { x / 9 } } { 9 } + \frac { e ^ { x / 9 } } { 81 } + C$&#10;B) $9 x e ^ { x / 9 } - 81 e ^ { x / 9 } + C$&#10;C) $\frac { x e ^ { x / 9 } } { 9 } - \frac { e ^ { x / 9 } } { 81 } + C$&#10;D) $9 x e ^ { x / 9 } + 81 e ^ { x / 9 } + C$

Accepted Answer

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

Question 5

Evaluate $\int x ( x - 2 ) ^ { 9 } d x$ .&#10;A) $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 5 x + 2 ) + C$&#10;B) $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( 5 x + 1 ) + C$&#10;C) $\frac { 1 } { 110 } ( x - 2 ) ^ { 10 } ( 10 x + 1 ) + C$&#10;D) $\frac { 1 } { 55 } ( x - 2 ) ^ { 10 } ( x + 10 ) + C$

Accepted Answer

Using integration by parts, let $ u = x $ and $ dv = (x-2)^9 dx $. Then $ du = dx $ and $ v = \frac{1}{10}(x-2)^{10} $. The integral becomes $\frac{1}{10}x(x-2)^{10} - \int \frac{1}{10}(x-2)^{10} dx$. Solving this gives the result in choice B.

Question 6

Evaluate $\int ( x - 3 ) \ln ( 2 x ) d x$ .&#10;A) $6 x - \frac { x ^ { 2 } } { 2 } + x ( x - 6 ) \ln ( 2 x )$ + C&#10;B) $\frac { ( x - 3 ) ^ { 2 } } { 2 } \ln ( 2 x )$ + C&#10;C) $\frac { x ^ { 2 } } { 2 } + x ( x - 6 ) \ln ( 2 x )$ + C&#10;D) $3 x - \frac { x ^ { 2 } } { 4 } + \frac { x ( x - 6 ) \ln ( 2 x ) } { 2 }$ + C

Accepted Answer

The answer of Evaluate \(\int ( x - 3 )...

Question 7

Evaluate the improper integral: $\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } } { \left( x ^ { 3 } + 7 \right) ^ { 2 } } d x$&#10;A) $\frac { 1 } { 104 }$&#10;B) $\frac { 1 } { 102 }$&#10;C) $\frac { 1 } { 68 }$&#10;D)102

Accepted Answer

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

Question 8

From time t = 0 to t = 4 an object's speed is given by the function $s ( t ) = t e ^ { 4 t }$ .Compute the distance travelled by the object during this time interval.Round your answer to two decimals.

A)8,330,724.81
B)8,330,728.68
C)8,330,738.94
D)8,330,830.01

Accepted Answer

To find the distance traveled by the object from $t = 0$ to $t = 4$, we need to integrate the speed function $s(t) = te^{4t}$ over this interval. The integral of $s(t)$ with respect to $t$ gives us the distance traveled.$$\text{Distance} = \int_{0}^{4} te^{4t} dt$$To solve this integral, we use integration by parts, where we let $u = t$ and $dv = e^{4t}dt$, which gives $du = dt$ and $v = \frac{1}{4}e^{4t}$.$$\begin{align*}\text{Distance} &= \left. \frac{1}{4}te^{4t} \right|_0^4 - \int_{0}^{4} \frac{1}{4}e^{4t} dt \&= \frac{1}{4} \cdot 4e^{16} - \left. \frac{1}{16}e^{4t} \right|_0^4 \&= e^{16} - \frac{1}{16}e^{16} + \frac{1}{16} \&= \frac{15}{16}e^{16} + \frac{1}{16} \&\approx 8,330,728.68\end{align*}$$Therefore, the distance traveled by the object during the time interval from $t = 0$ to $t = 4$ is approximately 8,330,728.68, which corresponds to choice B.

Question 9

Evaluate $\int x \ln \sqrt { x } d x$ .&#10;A) $\frac { x ^ { 2 } \ln x } { 4 } + C$&#10;B) $x ^ { 2 } \ln x + C$&#10;C) $\frac { x ^ { 2 } \ln x } { 4 } - \frac { x ^ { 2 } } { 8 }$ + C&#10;D)x + C

Accepted Answer

To solve the integral $\int x \ln \sqrt { x } d x$, we use integration by parts, where $u = \ln \sqrt{x} = \frac{1}{2} \ln x$ and $dv = x dx$. Differentiating $u$ gives $du = \frac{1}{2x} dx$, and integrating $dv$ gives $v = \frac{x^2}{2}$. Applying the integration by parts formula, $\int u dv = uv - \int v du$, we get $\frac{x^2}{2} \cdot \frac{1}{2} \ln x - \int \frac{x^2}{2} \cdot \frac{1}{2x} dx = \frac{x^2 \ln x}{4} - \int \frac{x}{4} dx = \frac{x^2 \ln x}{4} - \frac{x^2}{8} + C$, which matches option C.

Question 10

Given an initial population, $P _ { 0 } = 11,000$ ,a renewal rate,R = 100,and a survival function, $S ( t ) = t e ^ { - 0.3 t }$ ,with time t measured in years,determine the population at the end of 13 years.Round to two decimal places.

A)171.99
B)12,231.87
C)3,895.50
D)1,693.69

Accepted Answer

The answer of Given an initial population, \(P _ {...

Question 11

Find the average value of the function $f ( x ) = x e ^ { - x / 3 }$ over the interval from 0 to 4.Round to two decimal places.&#10;A)0.87&#10;B)3.46&#10;C)10.86&#10;D)7.79

Accepted Answer

The answer of Find the average value of the function...

Question 12

A hospital patient receives intravenously 15 units of a certain drug per hour.The drug is eliminated exponentially,so that the fraction that remains in the patients body for t hours is $f ( t ) = e ^ { - t / 6 }$ .If the treatment is continued indefinitely,approximately how many units of the drug will be in the patient's body in the long run? Round to two decimal places,if necessary.

A)90 units
B)2.50 units
C)75 units
D)21 units

Accepted Answer

The answer of A hospital patient receives intravenously 15 units...

Question 13

A manufacturer has found that marginal cost is $( 2 q + 1 ) e ^ { - 0.02 q }$ dollars per unit when q units have been produced.The total cost of producing the first 10 units is $100.What is the total cost of producing the first 30 units?

A)$545.39
B)$635.39
C)$1,124.68
D)$300.00

Accepted Answer

The answer of A manufacturer has found that marginal cost...

Question 14

Use integration by parts to evaluate the integral $\int \frac { x } { \sqrt { x + 16 } } d x$ .&#10;A) $\frac { 2 } { 3 } ( x + 16 ) ^ { 3 / 2 } ( x - 32 ) + C$&#10;B) $\frac { 2 } { 3 } ( x - 16 ) ^ { 1 / 2 } ( x + 32 ) + C$&#10;C) $\frac { 2 } { 3 } ( x + 16 ) ^ { 1 / 2 } ( x - 32 ) + C$&#10;D) $\frac { 3 } { 2 } ( x + 16 ) ^ { 1 / 2 } ( x - 32 ) + C$

Accepted Answer

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

Question 15

Evaluate $\int _ { 2 } ^ { \infty } \frac { 1 } { x ( \ln x ) ^ { 3 } } d x$ .&#10;A) $- \frac { 1 } { 2 ( \ln 2 ) ^ { 2 } }$&#10;B) $- \frac { 1 } { 2 \ln 2 }$&#10;C) $\frac { 1 } { 2 \ln 2 }$&#10;D) $\frac { 1 } { 2 ( \ln 2 ) ^ { 2 } }$

Accepted Answer

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

Question 16

After t hours on the job,a factory worker can produce $150 t e ^ { - 0.5 t }$ units per hour.How many units does the worker produce during the first 4 hours? Round to two decimal places.&#10;A)843.60 units&#10;B)397.00 units&#10;C)356.40 units&#10;D)243.60 units

Accepted Answer

The answer of After t hours on the job,a factory...

Question 17

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } x ^ { 3 } e ^ { - x ^ { 4 } n } d x$ Round to two decimal places,if necessary.&#10;A)1.75&#10;B)Undefined&#10;C)3.5&#10;D)2.63

Accepted Answer

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

Question 18

Demographic studies conducted in a certain city indicate that the fraction of the residents that will remain in the city for at least t years is $f ( t ) = e ^ { - t / 50 }$ .The current population of the city is 500,000,and it is estimated that new residents will be arriving at the rate of 1,000 people per year.If this estimate is correct,what will be the population in the long run?

A)40,800
B)50,000
C)550,000
D)40,500

Accepted Answer

The answer of Demographic studies conducted in a certain city...

Question 19

Assume a 6-year franchise is expected to generate profit at the rate of $\sqrt [ 3 ] { t ^ { 2 } + 19,000 }$ dollars per year.If,over the next 6 years,the prevailing annual interest rate remains fixed at 7%,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.

A)$1,123.47
B)$561.73
C)$374.49
D)$337.04

Accepted Answer

The answer of Assume a 6-year franchise is expected to...

Question 20

Use integration by parts to evaluate the integral $\int ( 15 - x ) e ^ { x } d x$ .&#10;A) $( 16 + x ) e ^ { x } + C$&#10;B) $( - 16 - x ) e ^ { x } + C$&#10;C) $( 16 - x ) e ^ { x } + C$&#10;D) $( 15 - x ) e ^ { x } + C$

Accepted Answer

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

Deck 6: Additional Topics in Integration