Calculus II: Master Integration Techniques, Sequences, and Series

Question 1

Calculate the integral of sin(x)cos(x) with respect to x.

Accepted Answer

1/2(sin(2x)) + C

Answer

1/2(sin^2(x) + cos^2(x)) + C

Answer

-1/2(sin(2x)) + C

Answer

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

Question 2

In the context of integration, which method involves systematically changing the variable of integration to simplify the integral?

Accepted Answer

Substitution

Answer

Integration by parts

Answer

Guess and check

Answer

Integration by trigonometric substitution

Question 3

Decompose the rational function ∫(x^2 + 2x + 1) / (x + 1)^3 dx using partial fractions, then evaluate the resulting integral.

Accepted Answer

x - 1 / (x + 1)^2 + 1 / (x + 1)^3

Answer

x + 1 / (x + 1)^2 - 1 / (x + 1)^3

Answer

x^2 + 2 / (x + 1)^2 + 1 / (x + 1)^3

Answer

-x + 1 / (x + 1)^2 + 1 / (x + 1)^3

Question 4

Find the indefinite integral of f(x) = x^3 + 2x^2 - 1 with respect to x.

Accepted Answer

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

Answer

x^4 + 2x^3 - x + C

Answer

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

Answer

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

Question 5

In Calculus II, which of the following is NOT a valid technique for finding the integral of a function?

Accepted Answer

Using trigonometric identities

Answer

Integration by substitution

Answer

Integration by parts

Answer

Integration using partial fractions

Question 6

To evaluate the integral ∫[(x^2 + 2x - 1) / (x - 1)] dx, which substitution is most appropriate?

Accepted Answer

u = x - 1

Answer

u = x^2 + 2x - 1

Answer

u = x^2

Answer

u = 2x - 1

Question 7

What is the indefinite integral of the function f(x) = x³ + 2x² - 5x + 1?

Accepted Answer

x⁴/4 + 2x³/3 - 5x²/2 + x + C

Answer

4x³ + 2x² - 5x + 1 + C

Answer

x³/3 + 2x²/2 - 5x/1 + 1 + C

Answer

x⁴ + 2x³ - 5x² + x + C

Question 8

Which of the following is not a valid method for calculating integrals?

Accepted Answer

Integration by Graphing

Answer

Integration by Substitution

Answer

Integration by Parts

Answer

Integration by Partial Fractions

Question 9

What is the Taylor series expansion for the function f(x) = e^x?

Accepted Answer

e^x = 1 + x + x^2/2 + x^3/3 + ...

Answer

e^x = 1 - x + x^2/2 - x^3/3 + ...

Answer

e^x = x + x^2/2 + x^3/3 + x^4/4 + ...

Answer

e^x = 1 + x + x^2/2 + x^3 + ...

Question 10

Which integration technique is specifically designed for trigonometric functions?

Accepted Answer

Trigonometric substitution

Answer

Substitution

Answer

Integration by parts

Answer

Partial fractions

Question 11

In Calculus II, which of these is NOT an integration technique?

Accepted Answer

Chain Rule

Answer

Integration by Substitution

Answer

Integration by Parts

Answer

Integration by Partial Fractions

Question 12

Calculate the indefinite integral of the function: ∫ (x² + 2x - 3) dx

Accepted Answer

x³/3 + x² - 3x + C

Answer

x²/2 + x - 3x + C

Answer

x³ + 2x² - 3x + C

Answer

2x³ + 4x² - 6x + C

Question 13

A particle's position is given by the parametric equations x = t^2 + 1 and y = 3t - 1. What is the velocity of the particle at time t?

Accepted Answer

The velocity is given by the derivatives of the parametric equations: dx/dt and dy/dt.

Answer

Velocity is the rate of change of displacement, so the velocity is √(dx/dt)^2 + (dy/dt)^2.

Answer

The velocity is equal to the sum of dx/dt and dy/dt.

Answer

Velocity is not defined for parametric equations.

Question 14

Convert the polar equation r = 4sin(θ) to rectangular coordinates.

Accepted Answer

(4cos(θ), 4sin(θ))

Answer

(4, 0)

Answer

(4, θ)

Question 15

A particle's acceleration along the x-axis is given by a(t) = sin(t) + cos(t). If its initial position and velocity are x(0) = 1 and v(0) = 2, respectively, what is its position at time t = π/2?

Accepted Answer

x(π/2) = π

Answer

x(π/2) = 1

Answer

x(π/2) = 2

Answer

x(π/2) = 2π

Question 16

Solve the differential equation y'' - 4y' + 4y = 3x^2 + 2sin(x) using the method of undetermined coefficients.

Accepted Answer

y = Ax^2 + Bx + C + Dcos(x) + Esin(x)

Answer

y = x^3 + cos(x)

Answer

y = e^(3x) + sin(2x)

Answer

y = x^2 + 2sin(x) + 3cos(x)

Question 17

Find the sum of the series ∑[n=1 to infinity] (1/n^2). The correct answer is π^2/6.

Accepted Answer

π^2/6

Answer

1

Answer

e

Answer

The series diverges

Question 18

Find the volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 4 about the x-axis. The correct answer is 16π/3.

Accepted Answer

16π/3

Answer

8π/3

Answer

32π/3

Answer

64π/3

Question 19

Find the surface area of the surface generated by rotating the curve y = x^3 from x = 0 to x = 1 about the x-axis. The correct answer is 6π.

Accepted Answer

6π

Answer

2π

Answer

4π

Answer

8π

Question 20

Determine whether the vector field F(x, y, z) = (y + z) i + (x + z) j + (x + y) k is conservative. If it is, find a potential function for it.

Accepted Answer

F is conservative, and a potential function is f(x, y, z) = xy + xz + yz.

Answer

F is conservative, and a potential function is f(x, y, z) = x^2 + y^2 + z^2.

Answer

F is not conservative.

Answer

F is conservative, but a potential function cannot be found.

Question 21

Determine the Taylor series expansion of f(x) = e^x around x = 0.

Accepted Answer

Σ(n=0 to ∞) (x^n / n!)

Answer

Σ(n=0 to ∞) (x^n / (n + 1)!)

Answer

Σ(n=0 to ∞) (x^n / 2^n)

Answer

Σ(n=0 to ∞) (x^n / n^2)

Question 22

Calculate the area under the curve of f(x) = x^3 - 2x^2 + 1 over the interval [0, 2].

Accepted Answer

6

Answer

2

Answer

4

Answer

8

Question 23

Evaluate the integral ∫(x^2 + 3x - 5) dx. Apply the correct method of integration.

Accepted Answer

(x^3/3 + 3x^2/2 - 5x) + C

Answer

(x^3 + 3x^2 - 5x) + C

Answer

(1/3x^3 + 3/2x^2 - 5x) + C

Answer

(3x^2/2 - 5x) + C

Question 24

Determine if the sequence {n/(n+1)} converges or diverges. Explain your reasoning.

Accepted Answer

Converges to 0

Answer

Converges to 1

Answer

Diverges

Answer

Converges to 2

Question 25

Use the chain rule to find the derivative of the function f(x) = (x^2 + 1)^5.

Accepted Answer

10x(x^2 + 1)^4

Answer

5x(x^2 + 1)^4

Answer

10(x^2 + 1)^4

Answer

5(x^2 + 1)^4

Question 26

Use Green's theorem to evaluate the line integral ∫C (x^2 + y^2) dx + (x - y) dy, where C is the boundary of the region enclosed by the curves y = x^2 and y = 4.

Accepted Answer

16/3

Answer

8/3

Answer

32/3

Answer

64/3

Question 27

Determine whether the series ∑ (n=1 to infinity) (1/n²) is convergent or divergent.

Accepted Answer

Convergent

Answer

Divergent

Answer

Cannot be determined

Question 28

Determine whether the series ∑ (n! / 10^n) converges or diverges.

Accepted Answer

Converges by the Ratio test

Answer

Diverges by the Ratio test

Answer

Cannot be determined

Question 29

Find the Taylor series representation for f(x) = sin(x) centered at x = 0.

Accepted Answer

∑ (-1)^n x^(2n+1) / (2n+1)!

Answer

∑ (-1)^n x^2n+1 / (2n+1)!

Answer

∑ x^(2n+1) / (2n+1)!

Answer

∑ x^2n+1 / (2n+1)!

Question 30

Find the general solution to the differential equation y'' + 4y' + 3y = sin(2x).

Accepted Answer

y(x) = c1 e^(-x) + c2 e^(-3x) + (x-1/3)sin(2x)

Answer

y(x) = c1 e^(x) + c2 e^(-3x) + (x+1/3)sin(2x)

Answer

y(x) = c1 e^(-x) + c2 e^(3x) + (x-1/3)sin(2x)

Answer

y(x) = c1 e^(x) + c2 e^(-3x) + (x+1/3)sin(2x)

Question 31

Find the Taylor series representation of f(x) = x * e^x centered at a = 0. Provide its radius and interval of convergence.

Accepted Answer

Taylor series: 1 + x + x^2/2! + x^3/3! + ...; Radius: ∞; Interval: (-∞, ∞)

Answer

Taylor series: x + x^2 + x^3 + x^4 + ...; Radius: 1; Interval: [-1, 1]

Answer

Taylor series: 1 + x + x^2/2!; Radius: 0; Interval: [0, 0]

Answer

Taylor series: x^2 + x^3 + x^4 + ...; Radius: 2; Interval: (-2, 2)

Question 32

Which of the following methods can be used to determine the convergence or divergence of an improper integral?

Accepted Answer

Finding the limit of the integrand as the upper or lower bound approaches infinity

Accepted Answer

Using the Comparison Test

Answer

Evaluating the definite integral over a finite interval

Question 33

Apply the Limit Comparison Test to determine if the series ∑[n=1 to ∞] (n^2 + 1) / (n^4 + 3n^2 + 2) converges or diverges. Which of the following series would be an appropriate comparison series?

Accepted Answer

∑[n=1 to ∞] 1/n^3

Answer

∑[n=1 to ∞] 1/n^2

Answer

∑[n=1 to ∞] 1/n^4

Question 34

Derive the Taylor series expansion of f(x) = ln(x) around a = 1.

Accepted Answer

∑(n=1 to infinity) [(-1)^(n-1) / n] (x - 1)^n

Answer

∑(n=1 to infinity) [(-1)^n(x - 1)^n]

Answer

∑(n=1 to infinity) [n(x - 1)^n]

Answer

∑(n=1 to infinity) [(1/n) x^n]

Question 35

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

Accepted Answer

x^2 + 3x - ln|x - 1| + C

Answer

x^2 - 2x + ln|x - 1| + C

Answer

(x - 1)^2 + C

Answer

Cannot be evaluated using indefinite integration techniques

Question 36

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

Accepted Answer

∑(x^n / n!)

Answer

∑((x - 0)^n / n!)

Answer

∑(x^n / n)

Answer

Cannot be represented as a Taylor series

Question 37

Use integration by substitution to evaluate ∫xcos(x^2) dx.

Accepted Answer

1/2sin(x^2) + C

Answer

x^2sin(x^2) + C

Answer

-1/2cos(x^2) + C

Question 38

Determine whether the series ∑(n^2 + 1) / (n^3 + 2n + 1) converges or diverges.

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined using the given information

Answer

Converges conditionally

Question 39

Determine whether the function f(x) = x^3 - 2x^2 + x - 2 is continuous at x = 1.

Accepted Answer

Continuous

Answer

Discontinuous

Answer

Cannot be determined

Answer

Discontinuous at x = 0

Question 40

Evaluate the limit lim (x→∞) (x^2 - 1)/(x^3 + 2x) using L'Hopital's rule.

Accepted Answer

0

Answer

∞

Answer

-1

Answer

1/∞

Question 41

Determine whether the improper integral ∫(0 to ∞) e^(-x^2) dx converges or diverges using integration by substitution and evaluating the resulting limit.

Accepted Answer

Converges

Answer

Diverges to -∞

Answer

Diverges to ∞

Answer

The integral is undefined

Question 42

Determine the Taylor series representation for the function f(x) = e^x about x = 0

Accepted Answer

∑(n=0 to ∞) (x^n)/n!

Answer

∑(n=0 to ∞) x^n

Answer

∑(n=1 to ∞) (x^n)/n

Answer

∑(n=1 to ∞) x^n/n

Question 43

Apply the Taylor series representation for cos(x) to approximate cos(π/3) up to the third order

Accepted Answer

1 - (π^2)/18 + (π^4)/216

Answer

-1 - (π^2)/18 + (π^4)/216

Answer

1 + (π^2)/18 - (π^4)/216

Question 44

What is the partial fraction decomposition of the rational expression $\frac{1}{x^2 - 4}$?

Accepted Answer

$\frac{1}{4(x - 2)} - \frac{1}{4(x + 2)}$

Answer

$\frac{1}{2(x - 2)} + \frac{1}{2(x + 2)}$

Answer

$\frac{1}{4(x - 2)} + \frac{1}{4(x + 2)}$

Answer

$\frac{1}{x - 2} - \frac{1}{x + 2}$

Question 45

Does the sequence $\{\frac{n}{n + 1}\}$ converge or diverge? If it converges, what is its limit?

Accepted Answer

Converges to 1

Answer

Diverges to infinity

Answer

Diverges

Answer

Converges to 0

Question 46

Evaluate the indefinite integral $\int \frac{x}{x^2 + 1} dx$.

Accepted Answer

$\frac{1}{2} \ln(x^2 + 1) + C$

Answer

$\ln(x^2 + 1) + C$

Answer

$\frac{1}{2} \ln(x) + C$

Answer

$\frac{x^2}{2} + C$

Question 47

Use integration by parts to evaluate the integral $\int x\sin(x) dx$.

Accepted Answer

$-x\cos(x) + \sin(x) + C$

Answer

$x\cos(x) + \sin(x) + C$

Answer

$-x\cos(x) - \sin(x) + C$

Answer

$x\sin(x) + \cos(x) + C$

Question 48

Find the partial fraction decomposition of $\frac{1}{(x-1)(x+2)}$.

Accepted Answer

$\frac{1}{3(x-1)} - \frac{1}{3(x+2)}$

Answer

$\frac{1}{x-1} - \frac{1}{x+2}$

Answer

$\frac{1}{x-1} + \frac{1}{x+2}$

Answer

$\frac{1}{3(x-1)} + \frac{1}{3(x+2)}$

Question 49

Does the infinite series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converge or diverge?

Accepted Answer

Converges

Answer

Converges conditionally

Answer

Diverges

Answer

Diverges to infinity

Question 50

What is the sum of the geometric series $\sum_{n=0}^{\infty} (\frac{1}{2})^n$?

Accepted Answer

2

Answer

1

Answer

$\frac{1}{2}$

Answer

$\frac{3}{2}$

Question 51

What is the interval of convergence for the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$?

Accepted Answer

(-∞, ∞)

Answer

(-1, 1)

Answer

(0, ∞)

Answer

[-1, 1]

Question 52

What is the Taylor series expansion of the function $e^x$ centered at $x=0$?

Accepted Answer

$\sum_{n=0}^{\infty} \frac{x^n}{n!}$

Answer

$\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$

Answer

$\sum_{n=0}^{\infty} x^n$

Answer

$\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$

Question 53

A curve is defined parametrically by the equations $x = t^2$ and $y = t^3$. Find the equation of the tangent line to the curve at the point (1, 1).

Accepted Answer

y = $\frac{3}{2}x - \frac{1}{2}$

Answer

y = x

Answer

y = 2x - 1

Answer

y = $\frac{1}{2}x + \frac{1}{2}$

Question 54

Find the arc length of the curve $y = \frac{x^2}{2}$ from $x = 0$ to $x = 1$.

Accepted Answer

$\frac{\sqrt{5}}{2} - \frac{1}{2}$

Answer

$\frac{\sqrt{3}}{2}$

Answer

$\frac{1}{2}$

Answer

$\frac{\sqrt{2}}{2}$

Question 55

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

Accepted Answer

x^2e^x - 2xe^x + 2e^x + C

Answer

x^2e^x + 2xe^x + C

Answer

2xe^x + e^x + C

Answer

(x^2 + 1)e^x + C

Question 56

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

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Answer

Converges conditionally

Question 57

Find the partial fraction decomposition of:  (2x + 1)/(x^2 - 5x + 6)

Accepted Answer

3/(x - 2) - 1/(x - 3)

Answer

1/(x - 2) - 2/(x - 3)

Answer

-3/(x - 2) + 1/(x - 3)

Answer

2/(x - 2) + 1/(x - 3)

Question 58

Calculate the integral:  ∫(sin^3(x) cos(x)) dx

Accepted Answer

(1/4)sin^4(x) + C

Answer

(1/3)cos^3(x) + C

Answer

-cos^4(x) + C

Question 59

A particle moves along a straight line with velocity v(t) = t^2 - 4t + 3 m/s. Find the total distance traveled by the particle from t = 0 to t = 3 seconds.

Accepted Answer

4 meters

Answer

3 meters

Answer

8/3 meters

Answer

16/3 meters

Question 60

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

Accepted Answer

(-∞, ∞)

Answer

(1, 3]

Answer

[1, 3)

Answer

(1, 3)

Question 61

Evaluate the improper integral:  ∫(1 to ∞) (1/x^2) dx

Accepted Answer

1

Answer

-1

Answer

∞

Answer

0

Question 62

Find the Taylor series expansion of f(x) = ln(x) centered at a = 1.

Accepted Answer

(x - 1) - (x - 1)^2/2 + (x - 1)^3/3 - ...

Answer

1 - (x - 1) + (x - 1)^2/2 - ...

Answer

1 + (x - 1) + (x - 1)^2/2 + ...

Answer

(x - 1)^2/2 - (x - 1)^3/3 + ...

Question 63

A tank is filled with 1000 liters of pure water. Brine containing 0.05 kg of salt per liter enters the tank at a rate of 10 liters per minute. The well-mixed solution drains from the tank at the same rate.  Set up a differential equation that models the amount of salt, A(t), in the tank at time t.

Accepted Answer

dA/dt = 0.5 - A/100

Answer

dA/dt = 0.5 + A/100

Answer

dA/dt = 5 + A/10

Answer

dA/dt = 5 - A/10

Question 64

Determine the arc length of the curve y = x^(3/2) from x = 0 to x = 4.

Accepted Answer

(8/27)(10√10 - 1)

Answer

(8/27)(√10 - 1)

Answer

(2/3)(10√10 - 1)

Answer

(2/3)(√10 - 1)