Question 1

Evaluate   dx.

Accepted Answer

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

Question 2

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

Accepted Answer

The answer of For what values of the constant k...

Question 3

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

Accepted Answer

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

Question 4

Evaluate   dx.

Accepted Answer

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

Question 5

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.

Accepted Answer

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

Question 6

If g(x) = A   + 3B   + 2Cx + D, where A, B, C, and D are constant real numbers, and if   is the Simpson's Rule approximation for the integral   dx, then   = 16B + 4D.

Accepted Answer

A) True 
 B)False

Question 7

Evaluate   dx.

Accepted Answer

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

Question 8

Use a change of variable to rewrite the improper integral I =   in a form suitable for numerical approximation using the Trapezoid Rule or Simpson's Rule.

Accepted Answer

Use x = @#IMG-DLM& ,

Question 9

Evaluate the integral   dx.

Accepted Answer

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 
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 10

Evaluate the integral   dx.

Accepted Answer

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

Question 11

Integrate   dx.

Accepted Answer

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

Question 12

Evaluate   dx.

Accepted Answer

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

Question 13

Suppose that   $\le$ 60 on the interval [0, 2] and that a Simpson's Rule approximation S2n for    dx based on 2n equal subintervals of [0, 2] has been calculated. What is the smallest interval you can be sure must contain I?

Accepted Answer

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

Question 14

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

Accepted Answer

Recall if x =tan(θ/2) , then s

Question 15

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

Accepted Answer

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 
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 16

Integrate   ln(5x) dx.

Accepted Answer

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

Question 17

Use Simpson's Rule with 4 and 8 subintervals to approximate I =   Give your answers to 5 decimal places. What are the actual errors in these approximations?

Accepted Answer

A)  S4 $\approx$ 2.00466, S8 $\approx$ 2.00029, I - S4 $\approx$ -0.00466, I - S8 $\approx$ -0.00029 
B)  S4 $\approx$ 2.00456, S8 $\approx$ 2.00027, I - S4 $\approx$ -0.00456, I - S8 $\approx$ -0.00027 
C)  S4 $\approx$ 2.00446, S8 $\approx$ 2.00024, I - S4 $\approx$ -0.00446, I - S8 $\approx$ -0.00024 
D)  S4 $\approx$ 2.00436, S8$\approx$ 2.00020, I - S4 $\approx$ -0.00436, I - S8 $\approx$ -0.00020 
E)  S4 $\approx$ 2.00476, S8 $\approx$ 2.00031, I - S4 $\approx$ -0.00476, I - S8 $\approx$ -0.00031 
A)  S4 $\approx$ 2.00466, S8 $\approx$ 2.00029, I - S4 $\approx$ -0.00466, I - S8 $\approx$ -0.00029 
B)  S4 $\approx$ 2.00456, S8 $\approx$ 2.00027, I - S4 $\approx$ -0.00456, I - S8 $\approx$ -0.00027 
C)  S4 $\approx$ 2.00446, S8 $\approx$ 2.00024, I - S4 $\approx$ -0.00446, I - S8 $\approx$ -0.00024 
D)  S4 $\approx$ 2.00436, S8$\approx$ 2.00020, I - S4 $\approx$ -0.00436, I - S8 $\approx$ -0.00020 
E)  S4 $\approx$ 2.00476, S8 $\approx$ 2.00031, I - S4 $\approx$ -0.00476, I - S8 $\approx$ -0.00031

Question 18

Evaluate, if convergent,   .

Accepted Answer

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

Question 19

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

Accepted Answer

The substitution x =

Question 20

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

Accepted Answer

A)  S8 $\approx$ 2.0468 
B)  S8 $\approx$ 2.0473 
C)  S8 $\approx$ 2.0477 
D)  S8 $\approx$ 2.0480 
E)  S8 $\approx$ 2.0465 
A)  S8 $\approx$ 2.0468 
B)  S8 $\approx$ 2.0473 
C)  S8 $\approx$ 2.0477 
D)  S8 $\approx$ 2.0480 
E)  S8 $\approx$ 2.0465

Evaluate dx.

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

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

Evaluate dx.

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.

If g(x) = A + 3B + 2Cx + D, where A, B, C, and D are constant real numbers, and if is the Simpson's Rule approximation for the integral dx, then = 16B + 4D.

Evaluate dx.

Use a change of variable to rewrite the improper integral I = in a form suitable for numerical approximation using the Trapezoid Rule or Simpson's Rule.

Evaluate the integral dx.

Evaluate the integral dx.

Integrate dx.

Evaluate dx.

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

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

Integrate ln(5x) dx.

Use Simpson's Rule with 4 and 8 subintervals to approximate I = Give your answers to 5 decimal places. What are the actual errors in these approximations?

Evaluate, if convergent, .

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

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

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

Exam 7: Techniques of Integration

Evaluate dx.

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

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

Evaluate dx.

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.

If g(x) = A + 3B + 2Cx + D, where A, B, C, and D are constant real numbers, and if is the Simpson's Rule approximation for the integral dx, then = 16B + 4D.

Evaluate dx.

Use a change of variable to rewrite the improper integral I = in a form suitable for numerical approximation using the Trapezoid Rule or Simpson's Rule.

Evaluate the integral dx.

Evaluate the integral dx.

Integrate dx.

Evaluate dx.

Suppose that ≤\le≤ 60 on the interval [0, 2] and that a Simpson's Rule approximation S2n for dx based on 2n equal subintervals of [0, 2] has been calculated. What is the smallest interval you can be sure must contain I?

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

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

Integrate ln(5x) dx.

Use Simpson's Rule with 4 and 8 subintervals to approximate I = Give your answers to 5 decimal places. What are the actual errors in these approximations?

Evaluate, if convergent, .

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

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

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

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

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