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Exam 13: Techniques of Integration

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Question 1

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Use the Trapezoidal Rule to approximate the value of the definite integral $\int _ { 0 } ^ { 3 } \sqrt { 1 + x } d x , n = 4$ . Round your answer to three decimal places.

A) 2.7931
B) 2.7955
C) 4.6552
D) 4.6615
E) 6.7643

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Question 2

Multiple Choice

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Use integration by parts to find the integral below. $\int 5 x ^ { n } \ln a x d x ( a \neq 0 , n \neq - 1 )$

A) $\frac { 5 x ^ { n } } { n } \ln a x - \frac { 5 } { n ^ { 2 } } x ^ { n } + C$
B) $\frac { 6 x ^ { n + 1 } } { n + 1 } \ln a x - \frac { 6 } { ( n + 1 ) ^ { 2 } } x ^ { n + 1 } + C$
C) $\frac { 5 x ^ { n } } { n } - \frac { 5 } { ( n + 1 ) ^ { 2 } } \ln a x + C$
D) $\frac { 5 x ^ { n + 1 } } { n + 1 } \ln a x - \frac { 5 } { ( n + 1 ) ^ { 2 } } x ^ { n + 1 } + C$
E) $\frac { 6 x ^ { n + 1 } } { n + 1 } - \frac { 6 } { n ^ { 2 } } \ln a x + C$

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D

Question 3

Multiple Choice

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Evaluate the definite integral $\int _ { 2 } ^ { 6 } \sqrt { 7 + x ^ { 2 } } d x$ . Round your answer to three decimal places.

Use the table of integrals to find the average value of the growth function $N = \frac { 330 } { 1 + e ^ { 5.7 - 0.25 t } }$ over the interval $[ 22,27 ]$ , where N the size of a population and t is the time in days. Round your answer to three decimal places.

Evaluate the improper integral if it converges, or state that it diverges. $\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 8 } } d x$

The probability of recall in an experiment is modeled by $P ( a \leq x \leq b ) = \int _ { a } ^ { b } \frac { 75 } { 14 } \left( \frac { x } { \sqrt { 4 + 5 x } } \right) d x , 0 \leq x \leq 1$ where x is the percent of recall. What is the probability of recalling between 50% and 70%? Round your answer to three decimal places.

The rate of change in the number of subscribers $S$ to a newly introduced magazine is modeled by $\frac { d S } { d t } = 1000 t ^ { 2 } e ^ { - 1 } , 0 \leq t \leq 6$ where $t$ is the time in years. Use Simpson's Rule $n = 12$ with to estimate the total increase in the number of subscribers during the first 6 years.

Approximate the value of the definite integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of n. Round your answers to three significant digits. $\int _ { 0 } ^ { 2 } e ^ { - x ^ { 2 } } d x , n = 4$

Use a table of integrals to find the indefinite integral $\int \frac { x ^ { 2 } } { ( 8 + 2 x ) ^ { 7 } } d x$ .

Present Value of a Continuous Stream of Income. An electronics company generates a continuous stream of income of $4 t$ million dollars per year, where t is the number of years that the company has been in operation. Find the present value of this stream of income over the first 9 years at a continuous interest rate of 12%. Round answer to one decimal place.

Use integration by parts to evaluate $\int 3 x ^ { 3 } \ln x d x$ .

Use a table of integrals with forms involving a + bu to find $\int \frac { x ^ { 2 } } { 8 + 11 x } d x$

Use a table of integrals to find the indefinite integral $\int x ^ { 9 } e ^ { x ^ { 10 } } d x$ .

Suppose the mean height of American women between the ages of 30 and 39 is 64.5 inches, and the standard deviation is 2.7 inches. Use a symbolic integration utility to approximate the probability that a 30-to 39-year-old woman chosen at random is between 5 feet 4 inches and 6 feet tall.

Find the indefinite integral. $\int t \sqrt { 4 t + 9 } d t$

Use integration by parts to find the integral below. $\int \ln x ^ { 3 } d x$

The capitalized cost $C$ of an asset is given by $C = C _ { 0 } + \int _ { 0 } ^ { n } C ( t ) e ^ { - r t } d t$ where $C _ { 0 }$ is the original investment, $t$ is the time in years, $r$ is the annual interest rate compounded continuously, and $C ( t )$ is the annual cost of maintenance (in dollars) . Find the capitalized cost of an asset (a) for 5 years, (b) for 10 years, and (c) forever. $C _ { 0 } = \$ 300,000 , C ( t ) = 15,000 , r = 6 \%$

Decide whether the integral is proper or improper. $\int _ { 0 } ^ { 5 } e ^ { - x } d x$

Identify u and dv for finding the integral using integration by parts. $\int x ^ { 4 } e ^ { 7 x } d x$

Evaluate the definite integral $\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x$ dx. Round your answer to three decimal places.