Question 1

Integrate   dx.

Accepted Answer

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

Question 2

Evaluate   dt
Hint: First use the substitution u =   .

Accepted Answer

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

Question 3

Integrate   .

Accepted Answer

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 
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 4

converges to - 2.

Accepted Answer

A) True 
 B)False

Question 5

Evaluate the integral   t dt.

Accepted Answer

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

Question 6

Evaluate   dx.

Accepted Answer

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

Question 7

Find the maximum value of    on [0, 1], where f(x) =   , and use it to obtain an upper bound for the absolute value of the error involved if the Trapezoid Rule approximation based on n equal subintervals is used to approximate I =   dx. How large should n be chosen to ensure that the error does not exceed   ?

Accepted Answer

A)    $\le$ 2 on [0, 1],   $\le$   , n = 10 will do 
B)    $\le$ 1 on [0, 1],   $\le$   , n = 8 will do 
C)    $\le$ 1 on [0, 1],   $\le$  , n = 8 will do 
D)    $\le$ 2 on [0, 1],   $\le$  , n = 9 will do 
E)    $\le$ 2 on [0, 1],  $\le$   , n = 15 will do 
A)    $\le$ 2 on [0, 1],   $\le$   , n = 10 will do 
B)    $\le$ 1 on [0, 1],   $\le$   , n = 8 will do 
C)    $\le$ 1 on [0, 1],   $\le$  , n = 8 will do 
D)    $\le$ 2 on [0, 1],   $\le$  , n = 9 will do 
E)    $\le$ 2 on [0, 1],  $\le$   , n = 15 will do

Question 8

Suppose that 0 $\le$   (x) $\le$ 3 on the interval [0, 2] and that a Trapezoid Rule approximation Tn for   based on n equal subintervals of [0, 2] has been calculated. Which is the interval you can be sure must contain I?

Accepted Answer

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

Question 9

Evaluate the integral   .

Accepted Answer

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

Question 10

Evaluate the Midpoint Rule approximation   for   dx. Round your answer to 4 decimal places.

Accepted Answer

A)  0.8475 
B)  0.9469 
C)  0.9480 
D)  0.9459 
E)  0.9445 
A)  0.8475 
B)  0.9469 
C)  0.9480 
D)  0.9459 
E)  0.9445

Question 11

Evaluate the integral   .

Accepted Answer

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

Question 12

Suppose that the six subintervals Simpson's Rule and the three subintervals Midpoint Rule approximations for the integral   dx are respectively given by S6 = 42 and M3 =36.Determine the Trapezoid Rule approximation   for the integral.

Accepted Answer

Recall S₆ = @#IMG-DLM& . Substituting the

Question 13

The integral   dx is convergent.

Accepted Answer

A) True 
 B)False

Question 14

Evaluate, if convergent,   .

Accepted Answer

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

Question 15

The following table gives values of an unknown function f(x) determined by experimental measurement. Find the best Midpoint Rule approximation you can for   dx based on the values given in the table.

Accepted Answer

A)  M4 = 2.0470 
B)  M4 = 2.0465 
C)  M4 = 2.0475 
D)  M4 = 2.0460 
E)  M4 = 2.0480 
A)  M4 = 2.0470 
B)  M4 = 2.0465 
C)  M4 = 2.0475 
D)  M4 = 2.0460 
E)  M4 = 2.0480

Question 16

The following table gives values of an unknown function f(x) determined by experimental measurement. Find the best Trapezoid Rule approximation you can for    dx based on the values given in the table.

Accepted Answer

A)  T8 = 2.0475 
B)  T8 = 2.0470 
C)  T8 = 2.0465 
D)  T8 = 2.0460 
E)  T8 = 2.0480 
A)  T8 = 2.0475 
B)  T8 = 2.0470 
C)  T8 = 2.0465 
D)  T8 = 2.0460 
E)  T8 = 2.0480

Question 17

Evaluate

Accepted Answer

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

Question 18

Evaluate   .

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

Evaluate

Accepted Answer

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

Integrate dx.

Evaluate dt Hint: First use the substitution u = .

Integrate .

converges to - 2.

Evaluate the integral t dt.

Evaluate dx.

Evaluate the integral .

Evaluate the Midpoint Rule approximation for dx. Round your answer to 4 decimal places.

Evaluate the integral .

Suppose that the six subintervals Simpson's Rule and the three subintervals Midpoint Rule approximations for the integral dx are respectively given by S₆ = 42 and M₃ =36.Determine the Trapezoid Rule approximation for the integral.

The integral dx is convergent.

Evaluate, if convergent, .

The following table gives values of an unknown function f(x) determined by experimental measurement. Find the best Midpoint Rule approximation you can for dx based on the values given in the table.

The following table gives values of an unknown function f(x) determined by experimental measurement. Find the best Trapezoid Rule approximation you can for dx based on the values given in the table.

Evaluate

Evaluate .

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

Evaluate

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Exam 7: Techniques of Integration

Integrate dx.

Evaluate dt Hint: First use the substitution u = .

Integrate .

converges to - 2.

Evaluate the integral t dt.

Evaluate dx.

Suppose that 0 ≤\le≤ (x) ≤\le≤ 3 on the interval [0, 2] and that a Trapezoid Rule approximation Tn for based on n equal subintervals of [0, 2] has been calculated. Which is the interval you can be sure must contain I?

Evaluate the integral .

Evaluate the Midpoint Rule approximation for dx. Round your answer to 4 decimal places.

Evaluate the integral .

Suppose that the six subintervals Simpson's Rule and the three subintervals Midpoint Rule approximations for the integral dx are respectively given by S6 = 42 and M3 =36.Determine the Trapezoid Rule approximation for the integral.

The integral dx is convergent.

Evaluate, if convergent, .

The following table gives values of an unknown function f(x) determined by experimental measurement. Find the best Midpoint Rule approximation you can for dx based on the values given in the table.

The following table gives values of an unknown function f(x) determined by experimental measurement. Find the best Trapezoid Rule approximation you can for dx based on the values given in the table.

Evaluate

Evaluate .

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

Evaluate

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Suppose that the six subintervals Simpson's Rule and the three subintervals Midpoint Rule approximations for the integral dx are respectively given by S₆ = 42 and M₃ =36.Determine the Trapezoid Rule approximation for the integral.

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