Question 1

Evaluate   dx.

Accepted Answer

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

Question 2

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).

Accepted Answer

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) =   
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 3

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

Accepted Answer

A)    < k <   
B)    < k <   
C)    < k <   
D)  k =   only 
E)    < k < $\infty$ 
A)    < k <   
B)    < k <   
C)    < k <   
D)  k =   only 
E)    < k < $\infty$

Question 4

Find an upper bound for the size of the error if the Trapezoidal Rule using 4 equal subintervals is used to approximate the integral   dx. Is the error positive or negative?

Accepted Answer

A)     0 
C)     0 
D)    <   , Error < 0 
E)    <   , Error < 0 
A)     0 
C)     0 
D)    <   , Error < 0 
E)    <   , Error < 0

Question 5

The integral I =   dx is improper and so unsuitable for numerical approximation by, say, the Trapezoid Rule or Simpson's Rule. Use a suitable change of variable to transform I into a proper integral these techniques can be applied to.

Accepted Answer

Let x = u³. The inter

Question 6

Evaluate   dx.

Accepted Answer

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

Question 7

Find the Midpoint Rule approximation   for I =   based on dividing [0, 1] into 5 equal subintervals. Quote your answer to 4 decimal places. Calculate the exact value of I and so determine the error in the approximation.

Accepted Answer

A)  M5 $\approx$ 0.7862, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$ - 0.0008 
B)  M5 $\approx$ 0.7862, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$ 0.0008 
C)  M5 $\approx$ 0.7872, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$ - 0.0018 
D)  M5 $\approx$ 0.7872, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$0.0008 
E)  M5 $\approx$ 0.7837, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$ 0.0017 
A)  M5 $\approx$ 0.7862, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$ - 0.0008 
B)  M5 $\approx$ 0.7862, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$ 0.0008 
C)  M5 $\approx$ 0.7872, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$ - 0.0018 
D)  M5 $\approx$ 0.7872, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$0.0008 
E)  M5 $\approx$ 0.7837, I = $\pi$/4 $\approx$ 0.7854, Error $\approx$ 0.0017

Question 8

The values of a continuous function f on the closed interval [2, 20] are provided in the table below:   
Use the table to find the Simpson's Rule approximation S6 for   dx.

Accepted Answer

Question 9

Integrate

Accepted Answer

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 
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 10

Evaluate   dx.

Accepted Answer

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

Question 11

Evaluate, if convergent,   .

Accepted Answer

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

Question 12

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

Accepted Answer

A)  0.9438 
B)  0.9443 
C)  0.9432 
D)  0.9450 
E)  0.9445 
A)  0.9438 
B)  0.9443 
C)  0.9432 
D)  0.9450 
E)  0.9445

Question 13

Integrate   dx.

Accepted Answer

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

Question 14

If g(x) is a polynomial of degree two, then the error involved in approximating the integral   using the Trapezoid Rule   is zero.

Accepted Answer

A) True 
 B)False

Question 15

Evaluate the integral   dx.

Accepted Answer

A)    ln   -   ln   + C 
B)    ln   -   ln   + C 
C)    ln   -   ln   + C 
D)  7 ln   - 3 ln   + C 
E)  7 ln   - 3 ln   + C 
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 16

Determine the exact error involved in approximating the integral   dx using the Simpson's Rule S20 .

Accepted Answer

A)  0.00032 
B)  0.06 
C)  0.00008 
D)  0 
E)  0.12 
A)  0.00032 
B)  0.06 
C)  0.00008 
D)  0 
E)  0.12

Question 17

Use Simpson's Rule with 8 subintervals to approximate I =   dx. Round your answer to 6 decimal places.

Accepted Answer

A)  S8 = 0.632121 
B)  S8 = 0.632120 
C)  S8 = 0.635423 
D)  S8 = 0.635427 
E)  S8 = 0.634215 
A)  S8 = 0.632121 
B)  S8 = 0.632120 
C)  S8 = 0.635423 
D)  S8 = 0.635427 
E)  S8 = 0.634215

Question 18

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

Accepted Answer

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

Question 19

Evaluate the integral   dx.

Accepted Answer

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

Question 20

Evaluate the integral   dx.

Accepted Answer

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

Evaluate dx.

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).

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

Find an upper bound for the size of the error if the Trapezoidal Rule using 4 equal subintervals is used to approximate the integral dx. Is the error positive or negative?

The integral I = dx is improper and so unsuitable for numerical approximation by, say, the Trapezoid Rule or Simpson's Rule. Use a suitable change of variable to transform I into a proper integral these techniques can be applied to.

Evaluate dx.

Find the Midpoint Rule approximation for I = based on dividing [0, 1] into 5 equal subintervals. Quote your answer to 4 decimal places. Calculate the exact value of I and so determine the error in the approximation.

The values of a continuous function f on the closed interval [2, 20] are provided in the table below: Use the table to find the Simpson's Rule approximation S₆ for dx.

Integrate

Evaluate dx.

Evaluate, if convergent, .

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

Integrate dx.

If g(x) is a polynomial of degree two, then the error involved in approximating the integral using the Trapezoid Rule is zero.

Evaluate the integral dx.

Determine the exact error involved in approximating the integral dx using the Simpson's Rule S₂₀ .

Use Simpson's Rule with 8 subintervals to approximate I = dx. Round your answer to 6 decimal places.

Evaluate the integral dx.

Evaluate the integral dx.

Preliminaries

Limits and Continuity

Differentiation

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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.

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).

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

Find an upper bound for the size of the error if the Trapezoidal Rule using 4 equal subintervals is used to approximate the integral dx. Is the error positive or negative?

The integral I = dx is improper and so unsuitable for numerical approximation by, say, the Trapezoid Rule or Simpson's Rule. Use a suitable change of variable to transform I into a proper integral these techniques can be applied to.

Evaluate dx.

Find the Midpoint Rule approximation for I = based on dividing [0, 1] into 5 equal subintervals. Quote your answer to 4 decimal places. Calculate the exact value of I and so determine the error in the approximation.

The values of a continuous function f on the closed interval [2, 20] are provided in the table below: Use the table to find the Simpson's Rule approximation S6 for dx.

Integrate

Evaluate dx.

Evaluate, if convergent, .

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

Integrate dx.

If g(x) is a polynomial of degree two, then the error involved in approximating the integral using the Trapezoid Rule is zero.

Evaluate the integral dx.

Determine the exact error involved in approximating the integral dx using the Simpson's Rule S20 .

Use Simpson's Rule with 8 subintervals to approximate I = dx. Round your answer to 6 decimal places.

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

Evaluate the integral dx.

Evaluate the integral dx.

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

The values of a continuous function f on the closed interval [2, 20] are provided in the table below: Use the table to find the Simpson's Rule approximation S₆ for dx.

Determine the exact error involved in approximating the integral dx using the Simpson's Rule S₂₀ .