Question 1

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

Accepted Answer

A)  $u = x ^ { 4 } ; d v = e ^ { 7 x } d x$ 
B)  $u = \int x ^ { 4 } ; d v = \int e ^ { 7 x } d x$ 
C)  $u = \int x ^ { 4 } d x , d v = e ^ { 7 x } d x$ 
D)  $u = \int x ^ { 4 } d x ; d v = \int e ^ { 7 x } d x$ 
E)  $u = x ^ { 4 } d x , d v = \int e ^ { 7 x } d x$ 
A)  $u = x ^ { 4 } ; d v = e ^ { 7 x } d x$ 
B)  $u = \int x ^ { 4 } ; d v = \int e ^ { 7 x } d x$ 
C)  $u = \int x ^ { 4 } d x , d v = e ^ { 7 x } d x$ 
D)  $u = \int x ^ { 4 } d x ; d v = \int e ^ { 7 x } d x$ 
E)  $u = x ^ { 4 } d x , d v = \int e ^ { 7 x } d x$ A

Question 2

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

Accepted Answer

A)  $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t + 3 ) } { 20 } + C$ 
B)  $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t - 6 ) } { 20 } + C$ 
C)  $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t - 3 ) } { 20 } + C$ 
D)  $\frac { ( 4 t - 9 ) ^ { 3 / 2 } ( 2 t + 3 ) } { 20 } + C$ 
E)  $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t + 6 ) } { 20 } + C$ 
A)  $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t + 3 ) } { 20 } + C$ 
B)  $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t - 6 ) } { 20 } + C$ 
C)  $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t - 3 ) } { 20 } + C$ 
D)  $\frac { ( 4 t - 9 ) ^ { 3 / 2 } ( 2 t + 3 ) } { 20 } + C$ 
E)  $\frac { ( 4 t + 9 ) ^ { 3 / 2 } ( 2 t + 6 ) } { 20 } + C$ C

Question 3

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

Accepted Answer

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)  $\frac { 6 x ^ { n + 1 } } { n + 1 } - \frac { 6 } { n ^ { 2 } } \ln a x + C$ 
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)  $\frac { 6 x ^ { n + 1 } } { n + 1 } - \frac { 6 } { n ^ { 2 } } \ln a x + C$ D

Question 4

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.

Accepted Answer

A) $143.7 million 
B) $81.6 million 
C) $182.7 million 
D) $343.2 million 
E) $85.8 million 
A) $143.7 million 
B) $81.6 million 
C) $182.7 million 
D) $343.2 million 
E) $85.8 million

Question 5

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

Accepted Answer

A)  $\ln x ^ { 4 } - 3 x ^ { 4 } + C$ 
B)  $x \ln x ^ { 4 } - 4 x ^ { 4 } + C$ 
C)  $\ln x ^ { 4 } - x ^ { 4 } + C$ 
D)  $x \ln x ^ { 3 } - 3 x + C$ 
E)  $4 x \ln x ^ { 4 } - 4 x + C$ 
A)  $\ln x ^ { 4 } - 3 x ^ { 4 } + C$ 
B)  $x \ln x ^ { 4 } - 4 x ^ { 4 } + C$ 
C)  $\ln x ^ { 4 } - x ^ { 4 } + C$ 
D)  $x \ln x ^ { 3 } - 3 x + C$ 
E)  $4 x \ln x ^ { 4 } - 4 x + C$

Question 6

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.

Accepted Answer

A) 0.8547 
B) 0.5707 
C) 0.4257 
D) 0.5734 
E) 0.9522 
A) 0.8547 
B) 0.5707 
C) 0.4257 
D) 0.5734 
E) 0.9522

Question 7

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.

Accepted Answer

A) 0.243 
B) 0.206 
C) 0.650 
D) 0.163 
E) 0.832 
A) 0.243 
B) 0.206 
C) 0.650 
D) 0.163 
E) 0.832

Question 8

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

Accepted Answer

A)  $\frac { 1 } { 9 }$ 
B)  $9$ 
C)  $8$ 
D)  $\frac { 1 } { 7 }$ 
E) diverges 
A)  $\frac { 1 } { 9 }$ 
B)  $9$ 
C)  $8$ 
D)  $\frac { 1 } { 7 }$ 
E) diverges

Question 9

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

Accepted Answer

A)  $\frac { - 3 x ^ { 4 } ( 4 \ln ( x ) + 1 ) } { 4 } + C$ 
B)  $\frac { 3 x ^ { 3 } ( 3 \ln ( x ) + 1 ) } { 3 } + C$ 
C)  $\frac { 3 x ^ { 4 } ( 4 \ln ( x ) + 1 ) } { 16 } + C$ 
D)  $\frac { 3 x ^ { 3 } ( 3 \ln ( x ) - 1 ) } { 9 } + C$ 
E)  $\frac { 3 x ^ { 4 } ( 4 \ln ( x ) - 1 ) } { 16 } + C$ 
A)  $\frac { - 3 x ^ { 4 } ( 4 \ln ( x ) + 1 ) } { 4 } + C$ 
B)  $\frac { 3 x ^ { 3 } ( 3 \ln ( x ) + 1 ) } { 3 } + C$ 
C)  $\frac { 3 x ^ { 4 } ( 4 \ln ( x ) + 1 ) } { 16 } + C$ 
D)  $\frac { 3 x ^ { 3 } ( 3 \ln ( x ) - 1 ) } { 9 } + C$ 
E)  $\frac { 3 x ^ { 4 } ( 4 \ln ( x ) - 1 ) } { 16 } + C$

Question 10

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.

Accepted Answer

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

Question 11

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

Accepted Answer

A)  $\frac { 1 } { 121 } ( 11 x - 8 \ln | 8 + 11 x | ) + C$ 
B)  $\frac { 1 } { 1331 } \left( 11 x - \frac { 64 } { 8 + 11 x } - 16 \ln | 8 + 11 x | \right) + C$ 
C)  $\frac { 1 } { 1331 } \left( \frac { 11 x } { 2 } ( 11 x - 16 ) + 64 \ln | 8 + 11 x | \right) + C$ 
D)  $\frac { 1 } { 121 } \left( \frac { 11 x } { 2 } ( 11 x - 16 ) + 64 \ln | 8 + 11 x | \right) + C$ 
E)  $\frac { 1 } { 121 } \left( 11 x - \frac { 64 } { 8 + 11 x } - 16 \ln | 8 + 11 x | \right) + C$ 
A)  $\frac { 1 } { 121 } ( 11 x - 8 \ln | 8 + 11 x | ) + C$ 
B)  $\frac { 1 } { 1331 } \left( 11 x - \frac { 64 } { 8 + 11 x } - 16 \ln | 8 + 11 x | \right) + C$ 
C)  $\frac { 1 } { 1331 } \left( \frac { 11 x } { 2 } ( 11 x - 16 ) + 64 \ln | 8 + 11 x | \right) + C$ 
D)  $\frac { 1 } { 121 } \left( \frac { 11 x } { 2 } ( 11 x - 16 ) + 64 \ln | 8 + 11 x | \right) + C$ 
E)  $\frac { 1 } { 121 } \left( 11 x - \frac { 64 } { 8 + 11 x } - 16 \ln | 8 + 11 x | \right) + C$

Question 12

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

Accepted Answer

A) 31.060 
B) 25.997 
C) 37.693 
D) 34.376 
E) 19.364 
A) 31.060 
B) 25.997 
C) 37.693 
D) 34.376 
E) 19.364

Question 13

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. $\int _ { 0 } ^ { 2 } e ^ { - x ^ { 2 } } d x , n = 4$

Accepted Answer

A) a. Trapezoidal Rule: $\approx 1.881$ b. Simpson's Rule: $\approx 0.882$ 
B) a. Trapezoidal Rule: $\approx 0.881$ b. Simpson's Rule: $\approx 0.882$ 
C) a. Trapezoidal Rule: $\approx 0.881$ b. Simpson's Rule: $\approx 1.882$ 
D) a. Trapezoidal Rule: $\approx 0.081$ b. Simpson's Rule: $\approx 0.882$ 
E) a. Trapezoidal Rule: $\approx 0.881$ b. Simpson's Rule: $\approx 0.082$ 
A) a. Trapezoidal Rule: $\approx 1.881$ b. Simpson's Rule: $\approx 0.882$ 
B) a. Trapezoidal Rule: $\approx 0.881$ b. Simpson's Rule: $\approx 0.882$ 
C) a. Trapezoidal Rule: $\approx 0.881$ b. Simpson's Rule: $\approx 1.882$ 
D) a. Trapezoidal Rule: $\approx 0.081$ b. Simpson's Rule: $\approx 0.882$ 
E) a. Trapezoidal Rule: $\approx 0.881$ b. Simpson's Rule: $\approx 0.082$

Question 14

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

Accepted Answer

A)  $\left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } + \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } - \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$ 
B)  $\frac { 1 } { 8 } \left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } + \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } - \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$ 
C)  $\frac { 1 } { 8 } \left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } + \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } + \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$ 
D)  $\frac { 1 } { 8 } \left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } - \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } + \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$ 
E)  $\left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } - \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } + \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$ 
A)  $\left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } + \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } - \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$ 
B)  $\frac { 1 } { 8 } \left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } + \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } - \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$ 
C)  $\frac { 1 } { 8 } \left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } + \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } + \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$ 
D)  $\frac { 1 } { 8 } \left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } - \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } + \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$ 
E)  $\left[ \frac { - 1 } { 4 ( 8 + 2 x ) ^ { 4 } } - \frac { 16 } { 5 ( 8 + 2 x ) ^ { 5 } } + \frac { 64 } { 6 ( 8 + 2 x ) ^ { 6 } } \right] + C$

Question 15

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.

Accepted Answer

A)  $\approx$ 1870 subscribers 
B)  $\approx$ 1780 subscribers 
C)  $\approx$ 1800 subscribers 
D)  $\approx$ 1878 subscribers 
E)  $\approx$ 1987 subscribers 
A)  $\approx$ 1870 subscribers 
B)  $\approx$ 1780 subscribers 
C)  $\approx$ 1800 subscribers 
D)  $\approx$ 1878 subscribers 
E)  $\approx$ 1987 subscribers

Question 16

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. $C _ { 0 } = \$ 300,000 , C ( t ) = 15,000 , r = 6 \%$

Accepted Answer

A) a. For $n = 5 , C$ $\approx$ $253,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $4,466,666.67 
B) a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $1,466,666.67 
C) a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $2807,922.43c. For $n = \infty , C$ $\approx$ $4,466,666.67 
D) a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $4,466,666.67 
E) a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $466,666.67 
A) a. For $n = 5 , C$ $\approx$ $253,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $4,466,666.67 
B) a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $1,466,666.67 
C) a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $2807,922.43c. For $n = \infty , C$ $\approx$ $4,466,666.67 
D) a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $4,466,666.67 
E) a. For $n = 5 , C$ $\approx$ $453,901.30b. For $n = 10 , C$ $\approx$ $807,922.43c. For $n = \infty , C$ $\approx$ $466,666.67

Question 17

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

Accepted Answer

A) 7.499 
B) 8.562 
C) 5.896 
D) 6.999 
E) 6.236 
A) 7.499 
B) 8.562 
C) 5.896 
D) 6.999 
E) 6.236

Question 18

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

Accepted Answer

A) The integral is improper. 
B) The integral is proper. 
A) The integral is improper. 
B) The integral is proper.

Question 19

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

Accepted Answer

A)  $\frac { 1 } { 9 } e ^ { x ^ { 10 } } + C$ 
B)  $\frac { 1 } { 10 } e ^ { x ^ { 9 } } + C$ 
C)  $\frac { 1 } { 10 } e ^ { x ^ { 10 } } + C$ 
D)  $\frac { 1 } { 9 } e ^ { x ^ { 9 } } + C$ 
E)  $\frac { 1 } { 10 } e ^ { x } + C$ 
A)  $\frac { 1 } { 9 } e ^ { x ^ { 10 } } + C$ 
B)  $\frac { 1 } { 10 } e ^ { x ^ { 9 } } + C$ 
C)  $\frac { 1 } { 10 } e ^ { x ^ { 10 } } + C$ 
D)  $\frac { 1 } { 9 } e ^ { x ^ { 9 } } + C$ 
E)  $\frac { 1 } { 10 } e ^ { x } + C$

Question 20

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.

Accepted Answer

A) 200.507 
B) 758.790 
C) 198.507 
D) 391.543 
E) 321.407 
A) 200.507 
B) 758.790 
C) 198.507 
D) 391.543 
E) 321.407

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

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

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

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

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.

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

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

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.

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

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

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 $\int \frac { x ^ { 2 } } { ( 8 + 2 x ) ^ { 7 } } d x$ .

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

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

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

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.

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Exam 13: Techniques of Integration

Identify u and dv for finding the integral using integration by parts. ∫x4e7xdx\int x ^ { 4 } e ^ { 7 x } d x∫x4e7xdx

Find the indefinite integral. ∫t4t+9dt\int t \sqrt { 4 t + 9 } d t∫t4t+9​dt

Use integration by parts to find the integral below. ∫5xnln⁡axdx(a≠0,n≠−1)\int 5 x ^ { n } \ln a x d x ( a \neq 0 , n \neq - 1 )∫5xnlnaxdx(a=0,n=−1)

Use integration by parts to find the integral below. ∫ln⁡x3dx\int \ln x ^ { 3 } d x∫lnx3dx

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.

Evaluate the improper integral if it converges, or state that it diverges. ∫1∞1x8dx\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 8 } } d x∫1∞​x81​dx

Use integration by parts to evaluate ∫3x3ln⁡xdx\int 3 x ^ { 3 } \ln x d x∫3x3lnxdx .

Use the Trapezoidal Rule to approximate the value of the definite integral ∫031+xdx,n=4\int _ { 0 } ^ { 3 } \sqrt { 1 + x } d x , n = 4∫03​1+x​dx,n=4 . Round your answer to three decimal places.

Use a table of integrals with forms involving a + bu to find ∫x28+11xdx\int \frac { x ^ { 2 } } { 8 + 11 x } d x∫8+11xx2​dx

Evaluate the definite integral ∫267+x2dx\int _ { 2 } ^ { 6 } \sqrt { 7 + x ^ { 2 } } d x∫26​7+x2​dx . Round your answer to three decimal places.

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. ∫02e−x2dx,n=4\int _ { 0 } ^ { 2 } e ^ { - x ^ { 2 } } d x , n = 4∫02​e−x2dx,n=4

Use a table of integrals to find the indefinite integral ∫x2(8+2x)7dx\int \frac { x ^ { 2 } } { ( 8 + 2 x ) ^ { 7 } } d x∫(8+2x)7x2​dx .

Evaluate the definite integral ∫13x2ln⁡x\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x∫13​x2lnx dx. Round your answer to three decimal places.

Decide whether the integral is proper or improper. ∫05e−xdx\int _ { 0 } ^ { 5 } e ^ { - x } d x∫05​e−xdx

Use a table of integrals to find the indefinite integral ∫x9ex10dx\int x ^ { 9 } e ^ { x ^ { 10 } } d x∫x9ex10dx .

Fundamental Concepts of Algebra

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Applications of the Derivative

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

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

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

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

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

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

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.

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

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

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 $\int \frac { x ^ { 2 } } { ( 8 + 2 x ) ^ { 7 } } d x$ .

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

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

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