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4

Which technique of integration is used to evaluate the integral of a product of two functions, often when integration by parts is required?

A. Trigonometric substitution

B. Integration by substitution

C. Integration by partial fractions

D. Integration by u-substitution

Correct Answer :

C. Integration by partial fractions


Related Questions

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4

What is a vector?

A. A point in space

B. A scalar quantity

C. A quantity with magnitude and direction

D. A constant value

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4

What is the volume of the region bounded by the surfaces z = x^2 + y^2 and z = 4 - x^2 - y^2?

A. 4p

B. 8p

C. 16p

D. 32p

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4

What is the derivative of ln(x) with respect to x?

A. 1/x

B. 1

C. x

D. 0

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4

If a car's speed is decreasing over time, what can you conclude about its acceleration?

A. The acceleration is positive.

B. The acceleration is negative.

C. The acceleration is zero.

D. Nothing about the acceleration can be determined.

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4

The solution to the logistic differential equation dy/dx = ky(1 - y) is often used to model population growth. What happens to the population when y = 1?

A. The population is increasing.

B. The population is decreasing.

C. The population is zero.

D. The population remains constant.

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4

Which test can be used to determine the convergence of a series when its terms are all positive?

A. Alternating Series Test

B. Ratio Test

C. Integral Test

D. Comparison Test

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4

A ball is thrown into the air, and its height (in meters) above the ground at time t (in seconds) is given by h(t) = 4.9t^2 + 15t + 2. What is its initial velocity?

A. 4.9 m/s

B. 15 m/s

C. 2 m/s

D. 0 m/s

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4

The cost C(x) of producing x units of a product is given by C(x) = 3x^2 + 2x + 7. What is the marginal cost when x = 5?

A. 47

B. 37

C. 27

D. 17

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4

What is the area under the curve y = sin(x) from x = 0 to x = p?

A. 0

B. 1

C. p

D. 2p

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4

What is the particular solution to the initial value problem dy/dx = 2x, y(0) = 1?

A. y = x^2 + 1

B. y = x^2

C. y = 2x + 1

D. y = 2x

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4

If f(x) = x^3 + 2x^2 - 5x + 1, what is f''(x)?

A. 3x^2 + 4x - 5

B. 6x + 4

C. 6x^2 + 4

D. 3x^2 + 4

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4

Which type of differential equation is represented by the equation dy/dx = k*y, where k is a constant?

A. Separable

B. Linear

C. Exact

D. Homogeneous

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4

What is the general solution to the differential equation dy/dx = 3x^2?

A. y = x^3 + C

B. y = 3x + C

C. y = x^3/3 + C

D. y = 3x^3 + C

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4

If a plane is defined by the equation 2x - 3y + z = 7, what is the normal vector to the plane?

A. <2, -3, 1>

B. <3, 2, -1>

C. <1, 3, -2>

D. <-2, 3, -1>

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4

What is the equation of the polar curve r = 2cos(?)?

A. A circle centered at the origin

B. A cardioid

C. A parabola

D. A line

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4

What is the area of the region enclosed by the polar curve r = 2sin(?)?

A. p square units

B. 4 square units

C. 2 square units

D. 0 square units

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4

The integral ?(0 to 1) x^3 dx is equal to:

A. 0

B. 1/4

C. 1/3

D. 1/2

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4

What is the derivative of the vector-valued function r(t) = <2t, t^2, 3t + 1> with respect to t?

A. <2, 2t, 3>

B. <2, 2t, 3t + 1>

C. <2t, 2t^2, 3>

D. <0, 0, 0>

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4

Which technique of integration is used to evaluate the integral of a product of two functions, often when integration by parts is required?

A. Trigonometric substitution

B. Integration by substitution

C. Integration by partial fractions

D. Integration by u-substitution

What is the correct answer?

4

What is the limit of (sin x) / x as x approaches 0?

A. 1

B. 0

C. -1

D. Undefined

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4

Which test is used to determine the convergence or divergence of an infinite series involving alternating terms?

A. Integral Test

B. Comparison Test

C. Alternating Series Test

D. Divergence Test

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4

Which method is commonly used to solve separable differential equations?

A. Laplace transforms

B. Substitution

C. Separation of variables

D. Variation of parameters

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4

What is the derivative of f(x) = 4x^3 - 2x^2 + 5x - 1 with respect to x?

A. 12x^2 - 4x + 5

B. 12x^2 - 4x - 5

C. 12x^2 - 2x + 5

D. 12x^2 - 2x - 5

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4

If f(x) = sin(x), what is f'(x)?

A. cos(x)

B. -cos(x)

C. sin(x)

D. -sin(x)

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4

What is the derivative of the constant function f(x) = 7?

A. 0

B. 1

C. 7

D. -7

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4

What is the directional derivative of the function f(x, y) = x^2 + y^2 at the point (1, 2) in the direction of the vector v = <3, 4>?

A. 1

B. 2

C. 3

D. 4

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4

The function P(t) represents the population of a city at time t. What does ?(a to b) P(t) dt represent?

A. The average population of the city on [a, b]

B. The change in population of the city from time a to time b

C. The rate of population growth at time a

D. The population of the city at time b

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4

Which of the following limits does not exist?

A. lim (x -> 0) (1/x)

B. lim (x -> 2) (x^2 - 4)/(x - 2)

C. lim (x -> 1) (x^2 - 1)/(x - 1)

D. lim (x -> 3) (1/x^2)

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4

What are the parametric equations for a circle of radius r centered at the origin in polar coordinates?

A. x = rcos(?), y = rsin(?)

B. x = rcos(t), y = rsin(t)

C. x = r?, y = r?

D. x = cos(?), y = sin(?)

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4

What is the antiderivative of e^(2x) with respect to x?

A. e^(2x) + C

B. (1/2)e^(2x) + C

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

D. (1/4)e^(2x) + C