Exam 6: Additional Topics in Integration

A hospital patient receives 4 units of a certain drug per hour intravenously. The drug is eliminated exponentially, so that the fraction that remains in the patients body for t hours is $f ( t ) = e ^ { - t / 12 }$ . If treatment is continued indefinitely, approximately how many units of the drug will be in the patient's body in the long run?

Evaluate $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 4 ] { x } } d x$

Evaluate $\int _ { 5} ^ { \infty } \frac { 1 } { x ^ { 2 } - x } d x$ . $\left( \text { Hint: } \frac { 1 } { x ^ { 2 } - x } = \frac { 1 } { x - 1 } - \frac { 1 } { x } \right)$

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 2 hours? Round to two decimal places

Given that $\int _ { 0 } ^ { \infty } e ^ { - x ^ { 2 } } d x = \frac { \sqrt { \pi } } { 2 }$ , evaluate $\int _ { 0 } ^ { \infty } e ^ { - 9 x ^ { 2 } } d x$ .

Evaluate $\int _ { 1 } ^ { \infty } \frac { 1 } { \sqrt [ 8 ] { x } } d x$

Assume a 7-year franchise is expected to generate profit at the rate of $\sqrt [ 3 ] { t ^ { 2 } + 18,000 }$ dollars per year. If, over the next 7 years, the prevailing annual interest rate remains fixed at 8%, 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.

Use integration by parts to evaluate the integral $\int ( 3 - x ) e ^ { x } d x$ .

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

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

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

A certain nuclear power plant produces radioactive waste at the rate of 500 pounds per year. The waste decays exponentially at the rate of 1.5% per year. How many pounds of radioactive waste from the plant will be present in the long run? Round to two decimal places, if necessary.

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.

After t hours on the job, a factory worker can produce $130 t e ^ { - 0.5 t }$ units per hour. How many units does the worker produce during the first 4 hours? Round to two decimal places.

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } \frac { 1 } { ( x - 7 ) \ln ( x - 7 ) } d x$ Round your answer to two decimal places, if necessary.

From time t = 0 to t = 2 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.

The long run capitalized cost of an asset that initially cost $C _ { 0 }$ dollars is given by $C = C _ { 0 } + \int _ { 0 } ^ { \infty } A ( t ) e ^ { - r t } d t$ where A(t) is the annual cost of maintenance and r is the annual rate of interest, compounded continuously. Find the long run capitalized cost, in dollars, in the case where $C _ { 0 } = \$ 5,000,000$ , A(t) = 5, 000(1 + 3t), and r = 0.08.

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

Evaluate $\int x ^ { 3 } \left( x ^ { 2 } - 3 \right) ^ { 7 } d x$ .

Given an initial population, $P _ { 0 } = 13,000$ , a renewal rate, R = 100, and a survival function, $S ( t ) = t e ^ { - 0.1 t }$ , with time t measured in years, determine the population at the end of 14 years. Round to two decimal places.