Calculus II: Master Integration, Applications, and Infinite Series

Question 1

Does the improper integral ∫(1/x) dx from x = 1 to x = ∞ converge or diverge?

Accepted Answer

Diverges

Answer

Converges

Answer

Cannot be determined

Answer

Converges to 0

Question 2

Find the indefinite integral of f(x) = sin(x)cos(x).

Accepted Answer

cos(2x)/2 + C

Answer

-cos(x)sin(x) + C

Answer

sin(x)cos(x) + C

Answer

-sin(x)cos(x) + C

Question 3

Determine whether the infinite series ∑(n=1 to ∞) ((-1)^n * n / (2n + 1)) converges or diverges. Explain your reasoning using the alternating series test.

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 4

Use integration by parts to calculate the integral ∫(x * sin(x)) dx. Clearly demonstrate each step of your solution.

Accepted Answer

∫(x * sin(x)) dx = x * cos(x) - sin(x) + C

Answer

∫(x * sin(x)) dx = x * cos(x) + sin(x) + C

Answer

∫(x * sin(x)) dx = x * sin(x) - cos(x) + C

Question 5

Determine the convergence behavior of the alternating series $displaystyle sum_{n=1}^infty (-1)^n / (2n + 1)$.

Accepted Answer

Conditionally convergent

Answer

Absolutely convergent

Answer

Divergent

Answer

Cannot be determined

Question 6

Which of the following is an example of an improper integral?

Accepted Answer

$displaystyle int_{-infty}^infty e^{-x^2} dx$

Answer

$displaystyle int_0^1 x dx$

Answer

$displaystyle int_0^{pi/2} sin(x) dx$

Answer

$displaystyle int_1^2 rac{1}{x-1} dx$

Question 7

Determine the radius of convergence of the series $displaystyle sum_{n=1}^infty rac{(x + 3)^n}{n}$.

Accepted Answer

3

Answer

1

Answer

$displaystyle infty$

Answer

Cannot be determined

Question 8

Which of the following is NOT an application of integrals in physics?

Accepted Answer

Determining the maximum height of a projectile

Answer

Calculating the area under a velocity-time graph to find displacement

Answer

Determining the volume of a solid of revolution

Answer

Finding the work done by a force over a distance

Question 9

Determine whether the series ∑(n=1 to ∞) (1/n^2) converges or diverges. Justify your answer.

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Answer

Converges conditionally

Question 10

Calculate the volume of the solid generated when the region bounded by the curves y = x^2 and y = 2x is rotated about the y-axis.

Accepted Answer

π ∫[0, 1] (4x^3 - x^5) dx

Answer

π ∫[0, 1] (2x^3 - x^5) dx

Answer

π ∫[0, 2] (4x^3 - x^5) dx

Answer

π ∫[0, 2] (2x^3 - x^5) dx

Question 11

Identify the defining characteristic of a convergent sequence.

Accepted Answer

The terms gradually approach a specific limit.

Answer

The terms continuously increase without bound.

Answer

The terms oscillate between two fixed values.

Answer

The terms remain constant throughout the sequence.

Question 12

Identify the correct integral for calculating the arclength of the curve y = ln(x) on the interval [1, e].

Accepted Answer

∫[1,e] √(1 + (1/x)^2) dx

Answer

∫[1,e] √(1 + (x)^2) dx

Answer

∫[1,e] √(x + 1) dx

Answer

∫[1,e] √(x - 1) dx

Question 13

Use the Limit Comparison Test to determine if the series ∑(n=1 to ∞) ((n + 2) / (n^3 + 5n^2 + 6)) converges or diverges.

Accepted Answer

Converges

Answer

Diverges

Answer

Inconclusive

Answer

More information needed

Question 14

Determine whether the sequence a_n = (n^2 + 3n + 2)/(2n^3 + 1) converges or diverges.

Accepted Answer

Convergent

Answer

Divergent

Answer

Cannot be determined

Question 15

Find the work done by the force F(x) = x^3 + 2x over the interval [0, 1].

Accepted Answer

3/4 joule

Answer

1 joule

Answer

4/3 joules

Answer

2 joules

Question 16

Find the convergence or divergence of the series ∑(n=1 to infinity) (n / (n^2 + 1)).

Accepted Answer

Converges by the Limit Comparison Test

Answer

Diverges by the Limit Comparison Test

Answer

Converges by the Ratio Test

Answer

Diverges by the Ratio Test

Question 17

Find the area bounded by the curves y = x and y = x^4 using integration.

Accepted Answer

1/12

Answer

1/6

Answer

1

Answer

1/2

Question 18

Calculate the definite integral: ∫[0, 1] (x^3 + 2x^2 - 3x + 1) dx

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 19

Determine whether the improper integral ∫[1, ∞) (1 / (x^2 + 1)) dx converges or diverges using the Direct Comparison Test.

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 20

Which of the following techniques is NOT used for integration?

Accepted Answer

Order of integration

Answer

U-substitution

Answer

Integration by parts

Answer

Partial fraction decomposition

Question 21

Consider the integral ∫(x^3 + 2x^2 - 5x + 1) dx. Which method of integration is most suitable?

Accepted Answer

U-substitution

Answer

Integration by parts

Answer

Partial fraction decomposition

Answer

Integration by trigonometric substitution

Question 22

Find the derivative of the function f(x) = ∫(t^2 + 1) dt from 0 to x.

Accepted Answer

2x

Answer

x^2 + 1

Answer

∫(t^2 + 1) dt

Answer

x

Question 23

Find the derivative of the function: f(x) = (x^2 + sin x)e^x

Accepted Answer

(x^2 + sin x + 2x)e^x

Answer

(x^2 + sin x)e^x

Answer

(x^2 + sin x - 2x)e^x

Answer

(x^2 + cos x + 2x)e^x

Question 24

Find the appropriate limits of integration for the volume of the solid generated by revolving the region bounded by y = x^2 and y = 4 about the x-axis.

Accepted Answer

0 to 2

Answer

-4 to 0

Answer

-2 to 2

Answer

0 to 4

Question 25

Find the polar form of the complex number z = 3 + 4i.

Accepted Answer

5(cos θ + i sin θ), where θ = arctan(4/3)

Answer

-5(cos θ - i sin θ), where θ = arctan(4/3)

Answer

-5(cos θ + i sin θ), where θ = arctan(4/3)

Answer

5(cos θ - i sin θ), where θ = arctan(4/3)

Question 26

Evaluate the integral ∫(x^2 + 1) / (x^4 + x^2 + 1) dx using the most appropriate technique.

Accepted Answer

Partial Fractions

Answer

Integration by Parts

Answer

Trigonometric Substitution

Answer

U-substitution

Question 27

Which of the following represents the general form of a power series?

Accepted Answer

∑ (n=0 to ∞) a_n (x - c)^n

Answer

∑ (n=1 to ∞) n a_n x^n

Answer

∫ (n=0 to ∞) a_n x^n dx

Question 28

Determine the area of the region bounded by the curves y = x^3, the x-axis, and the line x = 1.

Accepted Answer

4/3

Answer

8/3

Answer

2/3

Answer

1/3

Question 29

Find the derivative of the function f(x) = x * e^x.

Accepted Answer

e^x + x * e^x

Answer

x * e^(x-1)

Answer

e^x - x * e^x

Answer

2x * e^x

Question 30

Find the particular solution to the differential equation dy/dx + y = e^x that satisfies the initial condition y(0) = 0.

Accepted Answer

y = e^x - 1

Answer

y = -e^x + 1

Answer

y = e^x + 1

Question 31

Calculate the Maclaurin series for f(x) = sin x and state the interval of convergence.

Accepted Answer

x - x^3/3! + x^5/5! - x^7/7! + ...; (-∞, ∞)

Answer

sin x - cos x + x^2 - x^4 + ...; (0, ∞)

Answer

(x - 1) + (x - 1)^2 + (x - 1)^3 + ...; (1, ∞)

Question 32

Determine if the improper integral ∫(1 to ∞) 1/x^(1/2) dx converges or diverges, providing a justification.

Accepted Answer

Diverges

Answer

Cannot be determined

Answer

Converges

Answer

Alternates

Question 33

Evaluate the line integral ∫C (x^2 + y^2) ds along the curve C defined by y = x^2 from (0, 0) to (1, 1), justifying the use of the arc length parameterization.

Accepted Answer

16/15

Answer

8

Answer

16

Answer

4

Question 34

Find the volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 2 about the x-axis, utilizing the method of cylindrical shells.

Accepted Answer

32π/15

Answer

16π/15

Answer

8π/15

Answer

4π/15

Question 35

Which of the following is the indefinite integral of ∫(x^2 + sin(x)) dx?

Accepted Answer

x^3/3 - cos(x) + C

Answer

x^3/2 + cos(x) + C

Answer

x^4/4 + sin(x) + C

Answer

x^2/2 + cos(x) + C

Question 36

Find the area of the region bounded by the curves y = x^2 and y = 4.

Accepted Answer

16/3

Answer

4

Answer

32/3

Answer

8/3

Question 37

Which of the following is the Taylor series expansion of e^x centered at x = 0?

Accepted Answer

∑(n=0 to ∞) x^n / n!

Answer

∑(n=0 to ∞) n! x^n

Answer

∑(n=1 to ∞) x^n / n

Question 38

Find the radius of convergence of the series ∑(n=1 to ∞) (x^n) / n^2.

Accepted Answer

1

Answer

2

Answer

0

Question 39

Which of the following is the correct formula for integration by parts?

Accepted Answer

∫udv = uv - ∫vdu

Answer

∫vdu = uv - ∫udv

Answer

∫udv = uv +∫vdu

Answer

∫udv = uv + ∫vdu

Question 40

Find the derivative of the function f(x) = ∫(0 to x) sin(t^2) dt.

Accepted Answer

sin(x^2)

Answer

x sin(x^2)

Answer

x cos(x^2)

Answer

cos(x^2)

Question 41

Which of the following improper integrals is convergent?

Accepted Answer

∫(1 to ∞) 1 / x dx

Answer

∫(0 to 1) 1 / sqrt(x) dx

Answer

∫(-∞ to 0) e^x dx

Answer

∫(-∞ to ∞) x^2 dx

Question 42

Find the general solution of the differential equation y' + y = e^x.

Accepted Answer

y = e^x (C + ∫e^(-x) dx)

Answer

y = e^x (C - ∫e^(-x) dx)

Answer

y = e^(-x) (C - ∫e^x dx)

Answer

y = e^(-x) (C + ∫e^x dx)

Question 43

Which of the following sequences converges to 0?

Accepted Answer

a_n = 1 / n^2

Answer

a_n = 1 / n

Answer

a_n = n

Answer

a_n = ln(n)

Question 44

Find the limit of the function f(x) = (x^2 - 1) / (x - 1) as x approaches 1.

Accepted Answer

2

Answer

0

Answer

∞

Answer

1

Question 45

Find the volume of the solid generated by revolving the region bounded by the curves y = x^2 and y = 4 about the x-axis.

Accepted Answer

128π/5

Answer

64π/3

Answer

16π/3

Answer

32π/3

Question 46

Evaluate the improper integral ∫(1 to ∞) e^(-x) dx.

Accepted Answer

1

Answer

0

Answer

-1

Answer

∞

Question 47

Find the arc length of the curve y = ln(x) from x = 1 to x = e.

Accepted Answer

√(2)

Answer

√(e - 1)

Answer

e - 1

Answer

√(e)

Question 48

Determine the convergence or divergence of the series ∑(n=1 to ∞) 1/n^2.

Accepted Answer

Converges

Answer

Diverges

Answer

Diverges to ∞

Answer

Converges conditionally

Question 49

Find the Taylor series expansion of f(x) = sin(x) centered at x = 0.

Accepted Answer

∑(n=0 to ∞) (-1)^n * x^(2n+1) / (2n+1)!

Answer

∑(n=0 to ∞) x^(2n+1) / (2n)!

Answer

∑(n=0 to ∞) x^(2n) / (2n)!

Answer

∑(n=0 to ∞) (-1)^n * x^(2n) / (2n)!

Question 50

Determine the interval of convergence of the power series ∑(n=0 to ∞) (x - 2)^n / n!

Accepted Answer

(-∞, ∞)

Answer

[0, 4]

Answer

[1, 3]

Answer

(-1, 3)

Question 51

Find the work done in pumping all the water out of a cylindrical tank with radius 5 meters and height 10 meters, if the water level is initially at 8 meters.

Accepted Answer

1960000π J

Answer

3920000π J

Answer

7840000π J

Answer

980000π J

Question 52

Solve the differential equation dy/dx = y^2 * x.

Accepted Answer

y = -1/(x^2 + C)

Answer

y = 1/(x^2 + C)

Answer

y = x^2 + C

Answer

y = -x^2 + C

Question 53

Find the general solution of the second-order homogeneous differential equation y'' - 4y' + 4y = 0.

Accepted Answer

y = (C1 + C2x)e^(2x)

Answer

y = C1e^(2x) + C2xe^(2x)

Answer

y = C1e^(2x) + C2e^(-2x)

Question 54

Use the method of partial fractions to evaluate the integral ∫(1/(x^2 - 4)) dx.

Accepted Answer

(1/4) * ln(|(x - 2)/(x + 2)|) + C

Answer

ln(|(x - 2)(x + 2)|) + C

Answer

ln(|(x - 2)/(x + 2)|) + C

Answer

(1/2) * ln(|(x - 2)/(x + 2)|) + C

Question 55

Calculate the integral: ∫(x^3 + 2x) dx

Accepted Answer

(x^4)/4 + x^2 + C

Answer

(x^3)/3 + x + C

Answer

(x^4)/2 + x^2 + C

Answer

(x^2)/2 + x + C

Question 56

Find the area under the curve y = sin(x) from 0 to π/2.

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 57

Calculate the volume of the solid generated by rotating the region bounded by y = x and y = x^2 about the x-axis.

Accepted Answer

(π/2) * (1/3)

Answer

(π/2) * (1/2)

Answer

(π/2) * (1)

Answer

(π/2) * (2/3)

Question 58

Determine if the sequence 1, 2, 4, 8, 16,... is convergent or divergent.

Accepted Answer

Divergent

Answer

Convergent to 1

Answer

Convergent to 2

Answer

Convergent to 0

Question 59

Find the sum of the first 10 terms of the geometric series 2 + 6 + 18 + ...

Accepted Answer

2046

Answer

512

Answer

256

Answer

1022

Question 60

Determine the interval of convergence for the power series ∑(n=1 to infinity) ((-1)^n * x^n) / n^2

Accepted Answer

(-1, 1)

Answer

(-∞, ∞)

Answer

(-2, 2)

Answer

(0, 1)

Question 61

Find the Taylor series expansion of sin(x) about x = 0.

Accepted Answer

∑(n=0 to infinity) ((-1)^n * x^(2n+1)) / (2n+1)!

Answer

∑(n=1 to infinity) ((-1)^n * x^n) / n!

Answer

∑(n=1 to infinity) ((-1)^n * x^(2n+1)) / (2n+1)!

Answer

∑(n=0 to infinity) ((-1)^n * x^n) / n!

Question 62

Use the divergence test to determine whether the series ∑(n=1 to infinity) (1 / n^p) is convergent or divergent.

Accepted Answer

Convergent for p > 1, Divergent for p ≤ 1

Answer

Convergent for p < 1, Divergent for p ≥ 1

Answer

Divergent for p > 0, Convergent for p ≤ 0

Answer

Convergent for p > 0, Divergent for p ≤ 0

Question 63

Calculate the limit: lim(x->∞) (x - √(x^2 - 1)) / (x + √(x^2 - 1))

Accepted Answer

1

Answer

-1

Answer

∞

Answer

0

Question 64

Evaluate the integral: ∫(x^2 + 2x - 1) dx

Accepted Answer

(x^3)/3 + x^2 - x + C

Answer

(x^3)/3 + 2x^2 - x + C

Answer

(x^4)/4 + x^3 - x^2 + C