Exam 6: Additional Topics in Integration

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.

Evaluate $\int x e ^ { x y } d x$ .

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?

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } x ^ { 3 } e ^ { - x ^ { 4 } n } d x$ Round to two decimal places,if necessary.

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

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.

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.

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.

Evaluate $\int _ { 2 } ^ { \infty } \frac { 1 } { x ( \ln x ) ^ { 3 } } d x$ .

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?

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.

Evaluate $\int x \ln \sqrt { x } d x$ .

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

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

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.

Evaluate the improper integral: $\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } } { \left( x ^ { 3 } + 7 \right) ^ { 2 } } d x$

Use integration by parts to evaluate the integral $\int \frac { x } { \sqrt { x + 16 } } d x$ .

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.

Evaluate $\int _ { 1 } ^ { \infty } e ^ { - 6 x } d x$ .

Use integration by parts to evaluate the integral $\int ( 15 - x ) e ^ { x } d x$ .