Deck 7: Techniques of Integration

Question 1

Integrate dx.

A) x - + C
B) -x + + C
C) x + + C
D) x - + C
E) x + 2

Answer 1

x - + C

Question 2

Integrate dx.

A) x cos x - sin x + C
B) x sin x + cos x + C
C) -x cos x + sin x + C
D) x sin x - cos x + C
E) -x sin x + cos x + C

Answer 2

x sin x + cos x + C

Question 3

Find dx.

A) 1 -
B) -1
C) - 1
D)
E) -1 -

Answer 3

1 -

Question 4

Integrate dx.

A) ln x + + C
B) ln x - + C
C) ln x + x + C
D) ln x - x + C
E) ln x - + C

Question 5

Integrate dx.

A) +
B) -
C) -
D) +
E)

Question 6

Integrate ln(5x) dx.

A) ln(5x) - + C
B) ln(5x) + + C
C) ln(5x) + + C
D) ln(5x) - + C
E) ln(5x) - + C

Question 7

Evaluate .

A) ( ) + C
B) ( ) + + C
C) ( ) - + C
D) ( ) + ln(1 - ) + C
E) ( ) + 2x ( ) + C

Question 8

Integrate .

A) (2 - 2x - 1) + C
B) (2 + 2x - 1) + C
C) (2 - 2x + 1) + C
D) (2 + 2x + 1) + C
E) (2 - 2x + 1) + C

Question 9

Integrate dx.

A) - - + + C
B) - - + C
C) - + - + C
D) - - - + C
E) + - + C

Question 10

Evaluate the integral t dt.

A) +
B) -
C) -
D) +
E) +

Question 11

Integrate

A) x x - x + sin 2x + C
B) x x + x + sin 2x + C
C) x x - x - sin 2x + C
D) x x + x - sin 2x + C
E) x x - x + cos 2x + C

Question 12

Evaluate .

A) x (x) - ln(1 + ) + C
B) + C
C) x (x) - ln(1 + ) + C
D) arc + C
E) ln + C

Question 13

Evaluate the integral sin 4x dx.

A) (3 sin 4x - 4 cos 4x) + C
B) (4 sin 4x - 3 cos 4x) + C
C) (3 sin 4x + 4 cos 4x) + C
D) (4 sin 4x + 3 cos 4x) + C
E) (3 sin 4x - 4 cos 4x) + C

Question 14

Integrate dx.

A) 6 - 2e
B) 4e - 6
C) e + 3
D) 4e - 3
E) 2e - 1

Question 15

Evaluate the integral dx.

A) -
B) +
C) +
D) -
E)

Question 16

Integrate dx.

A) 6 - 2e
B) -
C) -
D) +
E) 2 + 6

Question 17

Integrate

A) ln x - + C
B) ln x + + C
C) ln x - + C
D) ln x + + C
E) ln x - + C

Question 18

Evaluate dx.

A) 8 + 4
B) 4
C) 8 -5
D) -4
E) 3

Question 19

Find a reduction formula for = and use it to evaluate I₃ = dx.

A) = x - n , - 3x + 6x ln x - 6x + C
B) = x + n , + 3x + 6x ln x + 6x + C
C) = x - , - x + x ln x - x + C
D) = x + , - x + x ln x - x + C
E) = x - n , - 3x - 6x ln x + 6x + C

Question 20

Find a reduction formula for I_n = and use it to evaluate I₄ = . dx

A) = - , = + C
B) = + , = + C
C) = - n , = + C
D) = + n , = + C
E) = + , = + C

Question 21

Let I_n = dx. Find a reduction formula for I_n in terms of I_n-2 valid for n $\le$ 3and use it to evaluate I₅ = dx.

A) = + , = + ln(1 + )
B) = - , = - ln(1 + )
C) = + , = + ln(1 + )
D) = + , = - ln(1 + )
E) = + , = + ln(1 + )

Question 22

Evaluate the integral dx.

A) + 3x - 9ln + C
B) + 3x + 9ln + C
C) - 3x + 9ln + C
D) + 3x + 9ln + C
E) + 3x + 3ln + C

Question 23

Evaluate the integral dx.

A) 4 ln - ln + C
B) 4 ln + ln + C
C) 4 ln + ln + C
D) 4 ln - ln + C
E) 2 ln - ln + C

Question 24

Evaluate the integral dx.

A) 2x + ln + C
B) 2x - ln + C
C) x - ln + C
D) x + ln + C
E) x - ln + C

Question 25

Evaluate the integral dx.

A) + - ln + ln + C
B) + + ln - ln + C
C) + - ln - ln + C
D) + + ln + ln + C
E) + - ln + ln + C

Question 26

Evaluate dx.

A) ln( - 1) +
B) ln( + 1) +
C) ln( - 1) -
D) ln( + 1) -
E) ln( ) -

Question 27

Evaluate the integral dx.

A) ln - ln + C
B) ln - ln + C
C) ln - ln + C
D) 7 ln - 3 ln + C
E) 7 ln - 3 ln + C

Question 28

Evaluate the integral dx.

A) - + 2x - 2 ln + C
B) - + 2x + 2 ln + C
C) + + 2x - 2 ln + C
D) + + 2x + 2 ln + C
E) + + 2x + 2 ln + C

Question 29

Evaluate the integral dx.

A) +
B) -
C) - +
D) - -
E) +

Question 30

Evaluate the integral dx.

A) - x - + 3 (x) + C
B) - x + + 3 (x) + C
C) - x + - 3 (x) + C
D) + x + + 3 (x) + C
E) - x - - 3 (x) + C

Question 31

Evaluate the integral dx.

A)
B)
C)
D)
E)

Question 32

A) (x) + ln( + 1) - x + C
B) ( + 1) (x) - x + C
C) + C
D) + C
E) (x) - ln( + 1) + x + C

Question 33

Evaluate dx.

A) ln + ln + + C
B) ln + ln + + C
C) ln + + C
D) ln + ln + + C
E) ln + ln -3ln + C

Question 34

Evaluate the integral .

A) + (2x) + C
B) - (2x) + C
C) - (2x) + C
D) + (2x) + C
E) + (2x) + C

Question 35

Evaluate dx.

A) 3x - ln( + 4) +12 + C
B) - + + C
C) + C
D) 3x - ln( + 4) + C
E) - + + C

Question 36

The correct form of the partial fraction decomposition for the function is given by

A) +
B) + +
C) +
D) +
E) + +

Question 37

Evaluate the integral .

A) ln + + C
B) ln - + C
C) ln - + C
D) ln + + C
E) ln + + C

Question 38

Evaluate .

A) ln - - + C
B) - ln - + + C
C) - ln - - + C
D) ln - + + C
E) ln - - + C

Question 39

Evaluate .

A) ln - ln + + C
B) - ln + ln + + C
C) - ln - ln + + C
D) ln + ln + + C
E) ln + ln + + C

Question 40

Evaluate dx.

A) 144
B) 121
C) 9 $\pi$
D) 124
E) -144

Question 41

Evaluate dx.

A) - + ln
B) + ln
C) - + ln
D) + ln
E) - - ln

Question 42

Evaluate dx.

A) + C
B) + C
C) + C
D) + C
E) + C

Question 43

Evaluate dx.

A) + a ln + C
B) - a ln + C
C) - a ln + C
D) + a ln + C
E) - ln + C

Question 44

Evaluate dx.

A) 16 $\pi$
B) 8 $\pi$
C)
D) 16 $\pi$ - 8
E) 12 $\pi$

Question 45

Evaluate dt
Hint: First use the substitution u = .

A) ln + C
B) + C
C) ( + 2) + C
D) + C
E) 2 + C

Question 46

Evaluate dx.

A) -2x + C
B) + C
C) + C
D) + C
E) - + C

Question 47

Evaluate

A) + C
B) - + C
C) + C
D) 3 + C
E) 3 + C

Question 48

Evaluate dx.

A) - + + C
B) - + + C
C) - + C
D) - - + C
E) + + C

Question 49

Evaluate

A) + C
B) + C
C) + C
D) + C
E) + C

Question 50

Let J = dx. The substitution x = tan( $\theta$ transforms the integral J into:

A)
B) 3
C) 3
D)
E) 3

Question 51

Use the half-angle substitution x = tan (θ/2) to evaluate dθ.

Question 52

Evaluate dx.

A) -
B) -
C) -
D) -
E) +

Question 53

What technique would you use to evaluate the integral I = Instead, try to evaluate it using Maple or another computer algebra system.

Question 54

What technique would you use to evaluate the integral I = Instead, try to evaluate it using Maple or another computer algebra system.

Question 55

Let F(x) = Use Maple or another computer algebra program to compute F(x) and an approximate value for F( ) correct to 5 decimal places.

A) F(x) = FresnelS ; F( ) $\approx$ 0.89480
B) F(x) = FresnelS ; F( ) $\approx$ 0.89483
C) F(x) = FresnelS ; F( ) $\approx$ 0.89486
D) F(x) = FresnelS ; F( ) $\approx$ 0.89489
E) F(x) = FresnelS ; F( ) $\approx$ 0.894878

Question 56

Let G(x) = dt. Use Maple or another computer algebra system to calculate G(1) correct to 5 decimal places, and also to calculate G(x).

A) G(1) $\approx$ 0.85562, G(x) =
B) G(1) $\approx$ 0.85558, G(x) =
C) G(1) $\approx$ 0.74682, G(x) =
D) G(1) $\approx$ 0.74685, G(x) =
E) G(1) $\approx$ 0.87649, G(x) =

Question 57

Evaluate the integral, dx.

A)
B) e
C) ln 3
D)
E) diverges to $\infty$

Question 58

Evaluate the integral

A) $\pi$ /2
B) $\pi$
C) 1/2
D) 1
E) divergent

Question 59

Evaluate the integral dx.

A) 2
B) 1
C) $\pi$
D) e
E) diverges to $\infty$

Question 60

Evaluate the improper integral dx or show it to diverges (to $\infty$ or $\infty$ ).

A) converges to 1 -
B) diverges to $\infty$
C) converged to - 1
D) diverges to - $\infty$
E) converges to

Question 61

Evaluate, if convergent, .

A) $\pi$
B) 2 $\pi$
C)
D)
E) divergent

Question 62

Evaluate, if convergent, cos x dx.

A)
B)
C) -
D) 0
E) divergent

Question 63

Evaluate the integral .

A) 3
B)
C)
D)
E) diverges to $\infty$

Question 64

Evaluate the integral .

A) 2
B)
C) 3
D) 4
E) diverges to $\infty$

Question 65

Evaluate

A) -
B) -
C)
D) -
E) diverges to - $\infty$

Question 66

Evaluate the improper integral dx or show it diverges (to $\infty$ or - $\infty$ ).

A) diverges to $\infty$
B) converges to - sin(3)
C) converges to 3 - sin(3)
D) diverges to - $\infty$
E) converges to sin(3) - 3cos(3)

Question 67

Evaluate, if convergent, dx.

A) 2 $\pi$
B) $\pi$
C) 1
D) 0
E) divergent

Question 68

Evaluate, if convergent, .

A) $\pi$
B) 1
C) 0
D)
E) diverges to $\infty$

Question 69

Which of the following is not an improper integral?

A) dx
B) dx
C) dx
D) dx
E) dx

Question 70

converges to - 2.

Question 71

Evaluate, if convergent, . dx

A) -2
B) -1
C) 2
D) 1
E) diverges to $\infty$

Question 72

Evaluate, if convergent, .

A)
B) -
C)
D) diverges to $\infty$
E) diverges to - $\infty$

Question 73

Evaluate, if convergent, .

A) 2
B) 2 $\pi$

C) 1
D) $\pi$

E) diverges to $\infty$

Question 74

Find the area under the curve y = and above the x-axis between x = -1 and x = 1.

A) 4 square units
B) 2 square units
C) 2 square units
D) 4 square units
E) diverges to $\infty$

Question 75

Find the area between the curves y = and y = to the right of x = 0 if the area is finite.

A) 3 square units
B) 3 square units
C) 2 square units
D) 2 square units
E) diverges to $\infty$

Question 76

Evaluate, if convergent, dx.

A) 2
B) 2 $\pi$
C) 5 $\pi$
D) 5
E) diverges to $\infty$

Question 77

Find, if finite, the area of the region lying between the graph of the function (x) and the line to the right of x = 0.

A) square units
B) $\pi$ square units
C) $\pi$ + 1 square units
D) 2 $\pi$ square units
E) diverges to $\infty$

Question 78

Evaluate, if convergent,

A)
B)
C)
D)
E) diverges to $\infty$

Question 79

Evaluate, if convergent, the improper integral

A)
B) 1
C) e
D)
E) diverges to $\infty$

Question 80

For what values of the constant k does the improper integral converge?