Deck 8: Techniques of Integration

Question 1

To evaluate the integral using Integration by Parts, we should choose:

A) .
B) .
C) .
D) .
E) .

Answer 1

.

Question 2

To evaluate the integral using integration by parts, the convenient choice is :

A) .
B) .
C) .
D) .
E) .

Answer 2

.

Question 3

Evaluate the integral using one or two methods, as indicated.

A) : the Substitution Method followed by Integration by Parts, twice.
B) : Integration by Parts followed by the Substitution Method.

Answer 3

A) B)

Question 4

Use the substitution and the trigonometric identity to evaluate the integral.

Question 5

Compute the integrals using the reduction formulas, or the substitution , and the identities , , as required.

A)
B)

Question 6

Compute the integral .

Question 7

Evaluate the integrals using Integration by Parts, the Substitution Method, or both methods.

A)
B)

Question 8

Compute the following integrals.

A)
B)

Question 9

For computing the integral , the most efficient method is

A) Integration by Parts with and .
B) substitution of followed by Integration by Parts.
C) Integration by Parts, twice.
D) Integration by Parts with and .
E) None of the methods covered so far is efficient in computing this integral.

Question 10

The value of the integral is:

A) .
B) .
C) .
D) .
E) .

Question 11

Use the substitution and the trigonometric identity , or reduction formulas as necessary, to calculate the integrals.

A)
B)

Question 12

Evaluate the integral .

A)
B)
C)
D)
E)

Question 13

Compute the integral .

Question 14

Evaluate the integrals using Integration by Parts, the Substitution Method, or both methods.

A)
B) . (Hint: Use .

Question 15

To evaluate the integral by Integration by Parts, the convenient choice is

A)
B)
C)
D)
E)

Question 16

Evaluate the integrals.

A)
B)

Question 17

To evaluate the integral using Integration by Parts, we should choose:

A) .
B) .
C) .
D) .
E) .

Question 18

Evaluate the integral

Question 19

Evaluate the integral .

Question 20

Which of the following methods is efficient in computing the integral ?

A) Substitution Method with
B) Integration by Parts with and
C) Integration by Parts with and
D) Integration by Parts with and
E) None of the techniques covered so far is efficient.

Question 21

Use the substitution and the identities to evaluate the integral .

Question 22

Use substitution and reduction formulas to evaluate the integral .

Question 23

Evaluate the integral .

Question 24

Evaluate the following integral using trigonometric identities and reduction formulas.

Question 25

Evaluate the following integral using trigonometric identities and reduction formulas.

Question 26

Use the substitution and the identity to evaluate the integral .

Question 27

Evaluate the integral .

Question 28

Evaluate the integral .

Question 29

Evaluate the integral .

Question 30

Calculate the following integral using reduction formulas as necessary.

Question 31

Evaluate the integral .

Question 32

Evaluate the integral .

Question 33

Evaluate the integral .

Question 34

Evaluate the integral .

Question 35

Calculate the following integrals using the reduction formulas when necessary.

A)
B)

Question 36

Evaluate the integral .

Question 37

Evaluate the integral .

Question 38

Evaluate the integral .

Question 39

Evaluate the integral .

Question 40

Calculate the integral .

Question 41

Evaluate the integral .

Question 42

Evaluate the integral .

Question 43

Evaluate the integral .

Question 44

The partial fraction decomposition of is:

A) .
B) .
C) .
D) .
E) .

Question 45

Evaluate the integral .

Question 46

The partial fraction decomposition of is:

A) .
B) .
C) .
D) .
E) .

Question 47

Evaluate the integral .

Question 48

Calculate the following integral in terms of inverse hyperbolic functions.

Question 49

Evaluate the integral .

Question 50

Use the substitution and the identities and to rewrite the following integrand as a rational function and evaluate it.

Question 51

Evaluate the integral .

Question 52

The partial fraction decomposition of is:

A) .
B) .
C) .
D) .
E) .

Question 53

Evaluate the following integrals.

A)
B)

Question 54

Evaluate the integral .

Question 55

Evaluate the integral .

Question 56

Calculate the following integral in terms of inverse hyperbolic functions.

Question 57

The partial fraction decomposition of is:

A) .
B) .
C) .
D) .
E) .

Question 58

Evaluate the following integrals.

A)
B)

Question 59

Evaluate .

Question 60

Use the substitution and the identities and to evaluate the integral .

Question 61

The time between customers at a checkout line is a random variable with exponential density. There is a 60% probability of waiting 1 min or more between customers. What is the average time between customers?

Question 62

Which two of the following integrals converge?

A)
B)
C)
D)

Question 63

Compute the integral ..

Question 64

Evaluate .

Question 65

The integral :

A) diverges since the power of is not less than 1.
B) converges since the power of is greater than 1.
C) diverges since .
D) converges since .
E) none of the above

Question 66

Which of the following integrals converges?

A)
B)
C)
D)
E)

Question 67

Evaluate the following improper integrals, if they exist.

A)
B)

Question 68

Verify that is a probability density function on and calculate its mean value.

Question 69

Which of the following integrals converges?

A)
B)
C)
D)
E) none of the above.

Question 70

Evaluate .

Question 71

To show that converges, we should use:

A) the Comparison Test and .
B) the Comparison Test and .
C) evaluation of the integral.
D) No method can be used since the function is not non-negative.
E) none of the above.

Question 72

Evaluate the following improper integrals.

A)
B)

Question 73

Find a constant such that is a probability density function on the interval and compute the probability .

Question 74

Compute the integral ..

Question 75

To show that converges, we should use:

A) the inequality and the Comparison Test.
B) the inequality and the Comparison Test.
C) evaluation of the integral.
D) No method can be used, since the function is not non-negative.
E) none of the above.

Question 76

Evaluate the following improper integrals.

A)
B)

Question 77

Evaluate the following improper integrals.

A)
B)

Question 78

Evaluate .

Question 79

To show that converges, we should use:

A) the inequality and the Comparison Test.
B) the inequality and the Comparison Test.
C) evaluation of the integral.
D) the inequality and the Comparison Test.
E) none of the above.

Question 80

Evaluate .