Deck 5: Integrals

Question 1

Use the Midpoint Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$

A)16.25
B)0.505
C)5.4167
D)1.5
E)5.625
F)11.25
G)5.3033
H)2.5

Answer 1

5.3033

Question 2

Use the Midpoint Rule with n = 4 to approximate the integral $\int _ { 1 } ^ { 3 } \frac { 1 } { x } d x$

A) $\frac { 3776 } { 3465 }$
B) $\frac { 7552 } { 3465 }$
C) $\frac { 1888 } { 3465 }$
D) $\frac { 7 } { 32 }$
E) $\frac { 5 } { 16 }$
F) $\frac { 5 } { 32 }$
G) $\frac { 7 } { 16 }$
H) $\frac { 5 } { 2 }$

Answer 2

$\frac { 3776 } { 3465 }$

Question 3

Use Simpson's Rule with n = 4 to approximate the integral $\int _ { 1 } ^ { 3 } \frac { 1 } { x } d x$

A) $\frac { 33 } { 10 }$
B) $\frac { 66 } { 10 }$
C) $\frac { 12 } { 10 }$
D) $\frac { 21 } { 20 }$
E) $\frac { 11 } { 10 }$
F) $\frac { 22 } { 10 }$
G) $\frac { 6 } { 10 }$
H) $\frac { 21 } { 10 }$

Answer 3

$\frac { 11 } { 10 }$

Question 4

Estimate using the Trapezoidal Rule with n = 4. Then use the error bound to estimate the accuracy.

Question 5

Use Simpson's Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$

A) $\frac { 65 } { 4 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 45 } { 8 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 65 } { 12 }$
F) $\frac { 45 } { 4 }$
G) $\frac { 7 } { 16 }$
H) $\frac { 5 } { 2 }$

Question 6

Consider the integral Approximating it by the Midpoint Rule with n equal subintervals, give an estimate for n which guarantees that the error is bounded by

Question 7

Use the Trapezoidal Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$

A) $\frac { 65 } { 4 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 45 } { 8 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 65 } { 12 }$
F) $\frac { 45 } { 4 }$
G) $\frac { 7 } { 16 }$
H) $\frac { 5 } { 2 }$

Question 8

Use the Trapezoidal Rule with n = 4 to approximate the integral $\int _ { 1 } ^ { 3 } \frac { 1 } { x } d x$

A) $\frac { 67 } { 30 }$
B) $\frac { 29 } { 10 }$
C) $\frac { 7 } { 8 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 32 }$
F) $\frac { 67 } { 60 }$
G) $\frac { 7 } { 16 }$
H) $\frac { 3 } { 4 }$

Question 9

Estimate using the Midpoint Rule with n = 4. Then use the error bound to estimate the accuracy.

Question 10

Use the Midpoint Rule with 2 equal subdivisions to get an approximation for ln 5.

Question 11

Estimate using Simpson's Rule with n = 4. Then use the error bound to estimate the accuracy.

Question 12

Use the Midpoint Rule with n = 2 to approximate the integral $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$

A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 7 } { 32 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 16 }$
F) $\frac { 5 } { 32 }$
G) $\frac { 7 } { 16 }$
H) $\frac { 5 } { 2 }$

Question 13

Suppose using n = 10 to approximate the integral of a certain function by the Trapezoidal Rule results in an upper bound for the error equal to $\frac { 1 } { 10 }$ . What will the upper bound become if we change to n = 20?

A) $\frac { 1 } { 10,000 }$
B) $\frac { 1 } { 100 }$
C) $\frac { 1 } { 80 }$
D) $\frac { 1 } { 160 }$
E) $\frac { 1 } { 1000 }$
F) $\frac { 1 } { 20 }$
G) $\frac { 1 } { 40 }$
H) $\frac { 1 } { 100,000 }$

Question 14

Use Simpson's Rule with n = 4 to approximate

Question 15

Use the Trapezoidal Rule with n = 2 to approximate the integral $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$

A) $\frac { 5 } { 16 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 5 } { 8 }$
E) $\frac { 1 } { 3 }$
F) $\frac { 7 } { 16 }$
G) $\frac { 2 } { 3 }$
H) $\frac { 3 } { 8 }$

Question 16

Suppose using n = 10 to approximate the integral of a certain function by Simpson's Rule results in an upper bound for the error equal to $\frac { 1 } { 10 }$ . What will the upper bound become if we change to n = 20?

A) $\frac { 1 } { 100 }$
B) $\frac { 1 } { 10,000 }$
C) $\frac { 1 } { 40 }$
D) $\frac { 1 } { 1000 }$
E) $\frac { 1 } { 160 }$
F) $\frac { 1 } { 20 }$
G) $\frac { 1 } { 100,000 }$
H) $\frac { 1 } { 80 }$

Question 17

Use Simpson's Rule with n = 2 to approximate the integral $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$

A) $\frac { 5 } { 8 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 3 } { 8 }$
D) $\frac { 2 } { 3 }$
E) $\frac { 7 } { 16 }$
F) $\frac { 1 } { 4 }$
G) $\frac { 9 } { 16 }$
H) $\frac { 1 } { 2 }$

Question 18

Use (a) the Trapezoidal Rule with n = 8 and (b) Simpson's Rule with n = 8 to approximate Round your answers to six decimal places.

Question 19

Use Simpson's Rule with n = 6 to approximate

Question 20

Use the Trapezoidal Rule with n = 1 to approximate the integral $\int _ { 0 } ^ { 1 } ( \sqrt { x } + 2 ) d x$

A) $\frac { 1 } { 2 }$
B) $\frac { 9 } { 16 }$
C) $\frac { 7 } { 16 }$
D) $\frac { 1 } { 4 }$
E) $\frac { 3 } { 8 }$
F) $\frac { 2 } { 3 }$
G) $\frac { 5 } { 2 }$
H) $\frac { 5 } { 8 }$

Question 21

Find the value of the integral $\int _ { 1 } ^ { 2 } \sqrt { u ^ { 2 } - 1 } d u$

A) $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 4 - \sqrt { 3 } )$
B) $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 2 + \sqrt { 3 } )$
C) $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 4 + \sqrt { 3 } )$
D) $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 2 - \sqrt { 3 } )$
E) $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 4 - \sqrt { 3 } )$
F) $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 4 + \sqrt { 3 } )$
G) $\sqrt { 3 } + \frac { 1 } { 2 } \ln ( 2 + \sqrt { 3 } )$
H) $\sqrt { 3 } - \frac { 1 } { 2 } \ln ( 2 - \sqrt { 3 } )$

Question 22

Two students use Simpson's Rule to estimate . One divides the interval into 30 equal subintervals and the other into 60 equal subintervals. How will the accuracy of their estimates compare?

Question 23

Evaluate $\int \frac { 3 x + 2 } { x - 1 } d x$

A) $5 \ln | x - 1 | + 3 x + C$
B) $5 \ln | x - 1 | + C$
C) $3 \ln | x - 1 | + 3 x + C$
D) $3 x + C$
E) $3 \ln | x - 1 | + 5 x + C$
F) $5 x + C$
G) $2 \ln | x - 1 | + 3 x + C$
H) $2 \ln | x | + C$

Question 24

Find the value of the integral $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x$

A)1
B) $\frac { \pi } { 16 }$
C) $\frac { \pi } { 4 }$
D)2
E) $\frac { \pi } { 32 }$
F) $\frac { \pi } { 8 }$
G) $\frac { \pi } { 64 }$
H) $\frac { 1 } { 6 }$

Question 25

Use the Table of Integrals in your textbook to evaluate each of the following:
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

Question 26

Find the partial fraction expansion of the rational function: $\frac { x } { x ^ { 2 } - 5 x + 6 }$ .

A) $\frac { 3 } { x - 3 } - \frac { 2 } { x - 2 }$
B) $\frac { 3 } { x - 3 } + \frac { 2 } { x - 2 }$
C) $\frac { 3 } { x + 3 } - \frac { 2 } { x - 2 }$
D) $\frac { 3 } { x - 3 } - \frac { 2 } { x + 2 }$
E) $\frac { 3 } { x + 3 } - \frac { 2 } { x + 2 }$
F) $\frac { - 3 } { x + 3 } - \frac { 2 } { x - 2 }$
G) $\frac { - 3 } { x - 3 } + \frac { 2 } { x - 2 }$
H) $\frac { - 3 } { x - 3 } + \frac { - 2 } { x - 2 }$

Question 27

(a) Estimate using Simpson's Rule with n = 4.(b) Estimate the error of the approximation in part (a).(c) How large should we take n to guarantee that the estimate by Simpson's Rule is accurate to within 0.001?

Question 28

The following table shows the speedometer readings of a truck, taken at ten minute intervals during one hour of a trip.Time (min)
0
10
20
30
40
50
60
Speed (mi/h)
40
45
50
60
70
65
60
Use the table and the indicated technique to estimate the distance that the truck traveled in the hour.(a) The Trapezoidal Rule
(b) The Midpoint Rule
(c) Simpson's Rule

Question 29

Evaluate $\int \frac { \cos x } { 4 + \sin ^ { 2 } x } d x$

A) $\tan ^ { - 1 } \left( \frac { \sin x } { 2 } \right) + C$
B) $\frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { \sin x } { 2 } \right) + C$
C) $\frac { 1 } { 2 } \tan ^ { - 1 } ( \sin x ) + C$
D) $\frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { \cos x } { 2 } \right) + C$
E) $\frac { 1 } { 4 } \tan ^ { - 1 } \left( \frac { \sin x } { 2 } \right) + C$
F) $\ln \left( \frac { \sin x } { 2 } \right) + C$
G) $2 \tan ^ { - 1 } \left( \frac { \sin x } { 2 } \right) + C$
H) $\frac { 1 } { 2 } \ln \left( \frac { \sin x } { 2 } \right) + C$

Question 30

Below is a table of values for a continuous function f. 0
0.5
1 2.0
1.5 0.5 (a) Use the Trapezoidal Rule with n = 4 to approximate (b) Use Simpson's Rule with n = 4 to approximate

Question 31

Use the Table of Integrals in your textbook to evaluate each of the following:
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

Question 32

Find the value of the integral $\int _ { 0 } ^ { x / 12 } \sin ^ { 2 } u d u$

A) $\frac { \pi - 3 } { 18 }$
B) $\frac { \pi - 3 } { 16 }$
C) $\frac { \pi - 3 } { 24 }$
D) $\frac { \pi - 3 } { 72 }$
E) $\frac { \pi - 3 } { 36 }$
F) $\frac { \pi - 3 } { 12 }$
G) $\frac { \pi - 3 } { 48 }$
H) $\frac { \pi - 3 } { 60 }$

Question 33

Intelligence Quotient (IQ) scores are assumed to be normally distributed in the population. The probability that a person selected at random from the general population will have an IQ between 100 and 120 is given by Use the graph of p (x) graphed below to answer the questions which follow: (a) Use Simpson's Rule with n = 4 to approximate (b) The probability that a person selected from the general population will have an IQ score between 80 and 120 is given by What is the approximate value of (c) Since p (x) represents a probability distribution, the entire area of the region under the graph is exactly 1. Using this information, what is the approximate probability that a person selected at random from the general population will have an IQ score over 120?

Question 34

Find the value of the integral $\int _ { 0 } ^ { x / 6 } \frac { \sin x } { \cos ^ { 3 } x } d x$

A)1
B) $- \frac { 2 } { 3 }$
C) $\frac { 2 } { 3 }$
D)2
E) $\frac { 1 } { 3 }$
F) $- \frac { 1 } { 3 }$
G) $\frac { 1 } { 6 }$
H) $- \frac { 1 } { 6 }$

Question 35

Find the value of the integral $\int _ { 0 } ^ { 1 } \sqrt { 1 - u ^ { 2 } } d u$

A)2
B)1
C) $\frac { \pi } { 4 }$
D) $\frac { 1 } { 4 }$
E) $\frac { 1 } { 2 }$
F) $\pi$
G) $\frac { \pi } { 2 }$
H) $2 \pi$

Question 36

Find the value of the integral $\int _ { 0 } ^ { x / 2 } \cos ^ { 3 } x d x$

A)1
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D)2
E)3
F) $\frac { \pi } { 2 }$
G) $\frac { 1 } { 6 }$
H) $\frac { \pi } { 2 } - \frac { \pi ^ { 3 } } { 24 }$

Question 37

Two students use Simpson's Rule to estimate . One divides the interval into 30 equal subintervals and the other into 60 equal subintervals. How will the accuracy of their estimates compare?

Question 38

The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpson's Rule to estimate the area of the pool.

Question 39

A scientist collects the following data and plots it in the coordinate plane. 2
2.5
3
3.5
4
4.5
5 4
10
8
6
14
10
12
(a) Use Simpson's rule with n = 6 to estimate the area under the graph of a continuous function drawn through these points.(b) If it is known that for all x, estimate the error involved in the approximation in part (a).(c) How large do we have to choose n so that the approximation (Simpson's Rule) to the integral is accurate to within 0.001?

Question 40

Find the value of the integral $\int _ { 0 } ^ { x / 8 } \sin ^ { 2 } ( 4 x ) d x$

A)1
B) $\frac { \pi } { 16 }$
C) $\frac { \pi } { 3 }$
D)2
E) $\frac { \pi } { 32 }$
F) $\frac { \pi } { 8 }$
G) $\frac { \pi } { 6 }$
H) $\frac { 1 } { 6 }$

Question 41

Find the value of the integral $\int _ { 0 } ^ { x / 2 } e ^ { x } \cos x d x$

A) $\frac { e ^ { s / 4 } + 1 } { 2 }$

B) $\frac { e ^ { s / 2 } + 1 } { 2 }$
C) $\frac { \pi } { 2 }$
D) $\frac { e ^ { x / 2 } - 1 } { 4 }$
E)2

F) $\frac { e ^ { x / 4 } - 1 } { 2 }$
11
G) $\frac { e ^ { x / 2 } + 1 } { 4 }$

H) $\frac { e ^ { x / 2 } - 1 } { 2 }$

Question 42

Find the value of the integral $\int _ { 1 } ^ { e } \ln x d x$

A) $e ^ { 2 }$
B) $e$
C)2
D) $e ^ { 2 } - e$
E) $e - 2$
F)1
G) $e - 1$
H) $\frac { e - 1 } { 2 }$

Question 43

Find the value of the integral $\int _ { 0 } ^ { x } x \cos x d x$

A) $- 2$
B) $2 \pi - 2$
C) $\frac { \pi } { 2 }$
D)4
E)2
F) $\pi$
G) $2 \pi$
H) $- 4$

Question 44

Find the value of the integral $\int _ { 0 } ^ { x / 2 } x ^ { 2 } \sin x d x$

A) $\pi - 2$

B) $\frac { \pi - 2 } { 4 }$
C) $\frac { \pi - 2 } { 2 }$
D)1
E) $2 - \pi$
F) $\frac { 2 - \pi } { 4 }$

G) $\frac { 2 - \pi } { 2 }$

H)0

Question 45

Evaluate the following integrals:
(a) (b) (c) (d)

Question 46

Find the partial fraction expansion of the rational function: $\frac { 9 } { ( x + 1 ) ^ { 2 } ( 2 - x ) }$ .

A) $\frac { - 1 } { x + 1 } + \frac { 3 } { ( x + 1 ) ^ { 2 } } + \frac { - 1 } { x - 2 }$
B) $\frac { 1 } { x + 1 } - \frac { 1 } { x - 2 }$
C) $\frac { - 1 } { x + 1 } + \frac { - 3 } { ( x + 1 ) ^ { 2 } } + \frac { - 1 } { x - 2 }$
D) $\frac { 3 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x - 2 }$
E) $\frac { 1 } { x + 1 } + \frac { 3 } { ( x + 1 ) ^ { 2 } } + \frac { 1 } { x - 2 }$
F) $\frac { 1 } { x + 1 } - \frac { 3 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x - 2 }$
G) $\frac { 1 } { x + 1 } + \frac { 3 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x - 2 }$
H) $\frac { - 1 } { x + 1 } + \frac { - 3 } { ( x + 1 ) ^ { 2 } } + \frac { 1 } { x - 2 }$

Question 47

Find the value of the integral $\int _ { 1 } ^ { 4 } \frac { \ln x } { x ^ { 2 } } d x$

A) $\theta$

B) $2 e - 1$
C) $\frac { 3 } { 2 } - \ln 2$
D) $\frac { 1 } { 2 } - \frac { \ln 2 } { 2 }$
E) $e - 2$
F) $e - 1$

G) $\frac { 3 } { 4 } - \frac { \ln 2 } { 2 }$

H) $1 - \ln 2$

Question 48

Evaluate the following integrals:
(a) (b) (c) (d)

Question 49

Find the value of the integral $\int _ { 0 } ^ { 1 } t ( 2 t - 1 ) ^ { 4 } d t$

A) $\frac { 1 } { 3 }$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 5 }$
D)1
E) $\frac { 1 } { 2 }$
F) $\frac { 1 } { 10 }$

G) $\frac { 1 } { 20 }$

H)0

Question 50

Find the value of the integral $\int _ { 0 } ^ { 1 } x \tan ^ { - 1 } x d x$

A) $\frac { \pi } { 4 }$

B) $\pi - 2$
C) $\frac { \pi } { 2 }$
D) $\frac { \pi - 2 } { 2 }$
E) $\frac { \pi - 2 } { 4 }$
F) $\pi - 1$

G) $\frac { \pi - 1 } { 2 }$

H) $\frac { \pi - 1 } { 4 }$

Question 51

Evaluate the following integrals:
(a) (b) (c) (d)

Question 52

Find the partial fraction expansion of the rational function: $\frac { x ^ { 2 } } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) }$ .

A) $\frac { 1 } { 2 ( x - 1 ) } + \frac { 1 } { 2 \left( x ^ { 2 } + 1 \right) }$
B) $\frac { 1 } { 2 ( x - 1 ) } + \frac { x } { 2 \left( x ^ { 2 } + 1 \right) }$
C) $\frac { 1 } { ( x - 1 ) } + \frac { x + 1 } { \left( x ^ { 2 } + 1 \right) }$
D) $\frac { 1 } { ( x - 1 ) } + \frac { 1 } { \left( x ^ { 2 } + 1 \right) }$
E) $\frac { 1 } { 2 ( x - 1 ) } + \frac { x + 1 } { 2 \left( x ^ { 2 } + 1 \right) }$
F) $\frac { - 1 } { 2 ( x - 1 ) } + \frac { x + 1 } { 2 \left( x ^ { 2 } + 1 \right) }$
G) $\frac { 1 } { ( x - 1 ) } + \frac { x } { \left( x ^ { 2 } + 1 \right) }$
H) $\frac { 1 } { 2 ( x - 1 ) } - \frac { x + 1 } { 2 \left( x ^ { 2 } + 1 \right) }$

Question 53

Evaluate $\int \frac { 1 } { x ^ { 2 } - 4 } d x$

A) $\frac { 1 } { 4 } \ln \left| \frac { x + 2 } { x - 2 } \right| + C$
B) $\ln \left| x ^ { 2 } - 4 \right| + C$
C) $4 \ln \left| \frac { x + 2 } { x - 2 } \right| + C$
D) $\frac { 1 } { 4 } \ln \left| \frac { x - 2 } { x + 2 } \right| + C$
E) $\frac { 1 } { 2 } \ln \left| \frac { x + 2 } { x - 2 } \right| + C$
F) $\frac { 1 } { 4 } \ln \left| x ^ { 2 } - 4 \right| + C$
G) $\frac { 1 } { 2 } \ln \left| \frac { x - 2 } { x + 2 } \right| + C$
H) $- \frac { 1 } { x } - 4 x + C$

Question 54

Find the value of the integral $\int _ { 0 } ^ { 1 } \arcsin t d t$

A) $\pi - 2$

B) $\frac { \pi - 2 } { 4 }$
C) $\frac { \pi - 2 } { 2 }$
D)1
E) $2 - \pi$
F) $\frac { 2 - \pi } { 4 }$

G) $\frac { 2 - \pi } { 2 }$
H)0

Question 55

Evaluate the following integrals:
(a) (b) (c) (d)

Question 56

Find the value of the integral $\int _ { 0 } ^ { 1 } \ln \left( 1 + x ^ { 2 } \right) d x$

A) $\ln 2$

B) $\frac { \pi } { 8 }$
C) $\frac { \pi } { 2 } - 2 + \ln 2$
D) $2 - \ln 2$
E) $\frac { \pi } { 4 } + \ln 2$
F) $\pi - 4$

G) $\pi - 2$

H) $\pi - \ln 2$

Question 57

Evaluate the following integrals:
(a) (b) (c) (d)

Question 58

Evaluate the following integrals:
(a) (b) (c) (d)

Question 59

Evaluate the following integrals:
(a) (b) (c) (d)

Question 60

Find the value of the integral $\int _ { 0 } ^ { 1 } x e ^ { x } d x .$

A)2
B) $e ^ { 2 } - e$
C)1
D) $e ^ { 2 }$
E) $e$
F) $e - 1$
G) $e - 2$
H) $\frac { e - 1 } { 2 }$

Question 61

Evaluate the integral $\int \cos \sqrt { x } d x$

A) $2 \sin \sqrt { x } + C$

B) $2 \sqrt { x } \cos \sqrt { x } + C$
C) $\sqrt { x } ( \cos \sqrt { x } + \sin \sqrt { x } ) + C$
D) $\frac { \cos \sqrt { x } + \sin \sqrt { x } } { \sqrt { x } } + C$
E) $2 ( \sqrt { x } \sin \sqrt { x } + \cos \sqrt { x } ) + C$
F) $2 ( \sqrt { x } \cos \sqrt { x } + \sin \sqrt { x } ) + C$

G) $\sqrt { x } \cos \sqrt { x } + \frac { \sin \sqrt { x } } { \sqrt { x } } + C$

H) $\sqrt { x } \sin \sqrt { x } + \frac { \cos \sqrt { x } } { \sqrt { x } } + C$

Question 62

Find the value of $\int _ { e } ^ { e ^ { 2 } } \frac { ( \ln x ) ^ { 2 } } { x } d x$

A) $\ln 2$
B) $\frac { 1 } { 2 } \ln 2$
C) $\frac { 1 } { 2 }$
D) $\frac { 3 } { 2 }$
E)1

F) $1 / ( \ln 2 )$

G)0
H)
$\frac { 7 } { 3 }$

Question 63

Find the value of the integral $\int _ { 0 } ^ { x / 2 } \sin ( 2 t ) d t$

A).0
B)1
C) $\frac { \sqrt { 3 } } { 2 }$
D) $\frac { \pi } { 2 }$
E) $\frac { \sqrt { 2 } } { 4 }$
F)2
G) $\frac { \sqrt { 2 } } { 2 }$

H) $\frac { 1 } { 2 }$

Question 64

Determine a reduction formula for

Question 65

Evaluate the following integrals:
(a) (b) (c) (d)

Question 66

Find the value of the integral $\int _ { 0 } ^ { x / 3 } \sec x \tan x ( 1 + \sec x ) d x$

A)4
B) $\frac { 5 } { 2 }$
C)3
D) $\frac { 11 } { 2 }$
E) $\frac { 9 } { 2 }$
F)2
G) $\frac { 7 } { 2 }$
H)5

Question 67

Find the value of the integral $\int _ { 0 } ^ { x / 4 } \sin ^ { 2 } x \cos x d x$

A) $\frac { \sqrt { 2 } } { 18 }$

B) $\frac { \pi } { 4 }$
C) $\frac { 3 \pi } { 2 }$
D) $\frac { \sqrt { 2 } } { 12 }$
E) $\frac { \sqrt { 2 } } { 6 }$
F) $\frac { \sqrt { 2 } } { 9 }$

G) $\frac { 2 \pi } { 3 }$

H) $\frac { \pi } { 2 }$

Question 68

(a) Use integration by parts to prove the reduction formula: (b) Demonstrate your understanding of this formula by using it to evaluate:

Question 69

Evaluate the following integrals:
(a) (b) (c) (d)

Question 70

Evaluate the following integrals:
(a) (b) (c) (d)

Question 71

Find the value of the integral $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( x ^ { 3 } + 1 \right) ^ { 2 } } d x$

A) $\frac { 3 } { 4 }$
B)2
C) $\frac { 3 } { 7 }$
D)
$\frac { 7 } { 3 }$
E) $\frac { 1 } { 6 }$
F) $\frac { 3 } { 2 }$
G) $\frac { 2 } { 3 }$
H)1

Question 72

Find the value of the integral $\int _ { 0 } ^ { x / 2 } \cos x \sin ( \sin x ) d x$

A) $\frac { \pi } { 2 }$

B) $1 - \frac { \pi } { 4 }$
C)sin 1
D)1 - cos 1
E) $\frac { \pi } { 2 } - \sin 1$
F) $\frac { \pi } { 4 } + \cos 1$
G) $1 + \frac { 3 \pi } { 4 }$
H)1 + tan 1

Question 73

Evaluate the integral $\int \frac { 1 + \ln x } { x \ln x } d x$

A) $\ln x + C$

B) $\ln \ln x + C$
C) $x + \ln x + C$
D) $\ln x + \ln \ln x + C$
E) $\frac { x } { \ln x } + C$
F) $\frac { \ln x } { ( x + \ln x ) } + C$

G) $x \ln x + C$

H) $x \ln \ln x + C$

Question 74

Find the value of the integral $\int _ { 0 } ^ { 1 } \left( x ^ { 3 } + 1 \right) ^ { 5 } x ^ { 2 } d x$

A)0
B)1
C) $\frac { 21 } { 2 }$

D) $\frac { 32 } { 9 }$
E) $\frac { 32 } { 3 }$
F)2
G) $\frac { 7 } { 2 }$
H) $\frac { 3 } { 2 }$

Question 75

Find the value of $\int _ { e } ^ { e } \frac { d x } { x \sqrt { \ln x } }$

A)0
B)1
C)2
D)3
E)4
F)5
G)6
H)7

Question 76

Evaluate the following integrals:
(a) (b) (c) (d)

Question 77

Let f be a twice differentiable function such that f(0) = 5, f(3) = 1, and (3) = . Determine the value of .

Question 78

Find the value of the integral $\int _ { 1 } ^ { 4 } \frac { 1 } { ( 1 + \sqrt { x } ) ^ { 2 } } \frac { 1 } { \sqrt { x } } d x$

A) $\frac { 6 } { 5 }$

B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 5 } { 2 }$
E) $\frac { 4 } { 9 }$
F) $\frac { 3 } { 2 }$

G) $\frac { 5 } { 6 }$

H) $\frac { 1 } { 6 }$

Question 79

Find the value of the integral $\int _ { - 3 } ^ { 3 } \frac { x } { \sqrt { 1 + 3 x ^ { 2 } } } d x$

A) $\frac { 2 } { 3 } ( \sqrt { 7 } - 1 )$

B) $\sqrt { 7 } - 1$
C)0
D) $\frac { 1 } { 3 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 1 } { \sqrt { 7 } }$
G) $\sqrt { 7 }$
H)Does not exist

Question 80

Evaluate $\int \cos ( \ln x ) d x$

A) $\sin ( \ln x ) + C$

B) $\frac { x ^ { 2 } } { 2 } [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$
C) $\cos \left( \frac { 1 } { x } \right) + C$
D) $\frac { x } { 4 } [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$
E) $\frac { \sin x } { x } + C$
F) $x [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$

G) $- \sin \left( \frac { 1 } { x } \right) + C$

H) $\frac { x } { 2 } [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$