Integrals: Definition, Rules, and Applications - Calculus for Liberal Arts

Question 1

Which of the following is the integral of (x^2 + 1)/(x^2 - 1) with respect to x:

Accepted Answer

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

Answer

1/2 ln(|x^2 + 1|) + C

Answer

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

Answer

ln(|x^2 + 1|) + C

Question 2

Which of the following is the correct substitution to evaluate the integral ∫sqrt(x^2 + 1) dx with respect to x:

Accepted Answer

u = sqrt(x^2 + 1)

Answer

u = x^2 + 1

Answer

u = x

Answer

u = 1/x

Question 3

Find the volume of the solid generated by rotating the region bounded by y = x^2 and y = 4 about the x-axis:

Accepted Answer

16/3 π

Answer

8/3 π

Answer

32/3 π

Answer

64/3 π

Question 4

Which of the following is the integral of the function f(x) = e^(2x) with respect to x:

Accepted Answer

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

Answer

e^(2x) + C

Answer

2e^(2x) + C

Answer

4e^(2x) + C

Question 5

Which of the following is the correct integral to find the work done by a force F(x) = x^2 over a distance of 2:

Accepted Answer

∫[0, 2] x^2 dx

Answer

∫[0, 1] x^2 dx

Answer

∫[1, 2] x^2 dx

Answer

∫[2, 0] x^2 dx

Question 6

Which of the following integrals represents the area under the continuous curve ( y = f(x) ) from ( x = a ) to ( x = b )?

Accepted Answer

( intlimits_a^b f(x)  dx )

Answer

( intlimits_b^a f(x)  dx )

Answer

( intlimits_0^a f(x)  dx + intlimits_a^b f(x)  dx )

Answer

( intlimits_0^b f(x)  dx )

Question 7

To find the integral of ( int cos(x)  dx ), which method of integration should be used?

Accepted Answer

Integration by Parts

Answer

Integration by Substitution (U-substitution)

Answer

Trigonometric Identities

Answer

Partial Fractions

Question 8

Obtain the indefinite integral of ( int rac{1}{x}  dx ):

Accepted Answer

( ln(|x|) + C )

Answer

( rac{1}{x} + C )

Answer

( x + C )

Answer

( e^x + C )

Question 9

Define the order of a differential equation:

Accepted Answer

The highest derivative that appears in the equation

Answer

The number of variables in the equation

Answer

The number of terms in the equation

Answer

The number of solutions to the equation

Question 10

Solve the differential equation ( rac{dy}{dx} = xy ):

Accepted Answer

( y = e^{rac{x^2}{2}} + C )

Answer

( y = x + C )

Answer

( y = rac{1}{x} + C )

Answer

( y = x^2 + C )

Question 11

Evaluate the convergence or divergence of the following improper integral: ∫[1, ∞) 1/x^2 dx

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Answer

Converges to 1

Question 12

Find the indefinite integral of the function f(x) = x^2.

Accepted Answer

x^3/3 + C

Answer

x^2/2 + C

Answer

x^4/4 + C

Answer

x/2 + C

Question 13

Which method is primarily used to determine the area under a curve?

Accepted Answer

Integration

Answer

Differentiation

Answer

Limits

Answer

Convergence tests

Question 14

Find the integral of the function h(x) = e^(2x) with respect to x.

Accepted Answer

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

Answer

e^(2x) + C

Answer

2e^(2x) + C

Answer

e^(x^2) + C

Question 15

Calculate the integral that represents the volume of a solid generated by rotating the region bounded by the curves y = x^2 and y = 4 about the x-axis.

Accepted Answer

∫[0, 2] π(16 - x^2) dx

Answer

∫[0, 2] π(x^2 - 16) dx

Answer

∫[0, 2] π(x^2 + 16) dx

Answer

∫[0, 2] π(16 - 4x^2) dx

Question 16

Which of the following is the formula for the substitution rule for indefinite integrals?

Accepted Answer

∫[a, b] f(g(x)) * g'(x) dx = ∫[g(a), g(b)] f(u) du, where u = g(x).

Answer

∫[a, b] f(g(x)) * g'(x) dx = ∫[g(a), g(b)] f(u) du, where u = x.

Answer

∫[a, b] f(g(x)) * g'(x) dx = ∫[a, b] f(u) du, where u = g(x).

Answer

∫[a, b] f(g(x)) * g'(x) dx = ∫[g(a), g(b)] f(x) dx, where u = g(x).

Question 17

What is the indefinite integral of e^x?

Accepted Answer

e^x + C

Answer

-e^x + C

Answer

1/e^x + C

Answer

-1/e^x + C

Question 18

What is the indefinite integral of 1/x?

Accepted Answer

ln(|x|) + C

Answer

-ln(|x|) + C

Answer

1/ln(x) + C

Answer

-1/ln(x) + C

Question 19

According to the power rule for indefinite integrals, what is the antiderivative of x^n, where n is a positive integer?

Accepted Answer

x^(n+1)/(n+1) + C

Answer

-x^(n+1)/(n+1) + C

Answer

(n+1)x^n + C

Answer

-(n+1)x^n + C

Question 20

If the velocity of an object is given by the function v(t), which integral represents the total distance traveled by the object over the time interval [a, b]?

Accepted Answer

∫[a, b] v(t) dt

Answer

∫[a, b] v^2(t) dt

Answer

∫[a, b] 1/v(t) dt

Answer

∫[a, b] t*v(t) dt

Question 21

In physics, which of the following integrals represents the work done by a constant force F acting on an object over a displacement x?

Accepted Answer

∫[0, x] F dx

Answer

∫[0, x] F/x dx

Answer

∫[0, x] x*F dx

Answer

∫[0, x] 1/F dx

Question 22

Clarified: Which of the following accurately defines an integral?

Accepted Answer

The area enclosed underneath the curve of a function.

Answer

The volume of a three-dimensional object.

Answer

The rate of change of a function.

Answer

The length of a curve traced by a function.

Question 23

Simplified: Which integration rule is used to find the integral of the product of two functions?

Accepted Answer

Integration by Parts

Answer

Product Rule

Answer

Chain Rule

Answer

u-Substitution

Question 24

Simplified: Integrate the function ∫[(x^2 + 1) / (x - 1)] dx.

Accepted Answer

x - ln|x - 1| + C

Answer

x + ln|x - 1| + C

Answer

x^2 - x + C

Answer

x^2 + x + C

Question 25

Find the integral of the function: ∫(x^2 + 3x - 2) dx

Accepted Answer

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

Answer

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

Answer

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

Answer

x^5 + 3x^4 - 2x^3 + C

Question 26

Evaluate the indefinite integral: ∫√(x) dx

Accepted Answer

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

Answer

2 * x^(1/2) + C

Answer

x^(3/2) + C

Answer

3 * x^(1/2) + C

Question 27

Which substitution is appropriate for integrating ∫(x + 1)e^(x^2 + 2x) dx?

Accepted Answer

u = x^2 + 2x

Answer

u = x + 1

Answer

u = e^(x^2 + 2x)

Answer

u = x

Question 28

Determine the area under the curve y = x^3 - 2x^2 + 1 between x = 0 and x = 2.

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 29

Simplify the integral: ∫(2cos(x) - 3sin(x)) dx

Accepted Answer

-2sin(x) + 3cos(x) + C

Answer

2sin(x) + 3cos(x) + C

Answer

2cos(x) - 3sin(x) + C

Answer

2sin(x) - 3cos(x) + C

Question 30

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

Accepted Answer

e^x + (x^3)/3 + C

Answer

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

Answer

e^x + x^3 + C

Answer

e^x - x^3 + C

Question 31

Which statement describes the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

For a continuous function f(x) on [a, b], ∫[a, b] f(x) dx = F(b) - F(a), where F'(x) = f(x)

Answer

For a differentiable function f(x) on [a, b], ∫[a, b] f'(x) dx = f(b) - f(a)

Answer

For a continuous function f(x) on [a, b], there exists a number c in [a, b] such that ∫[a, b] f(x) dx = f(c) * (b - a)

Answer

For a differentiable function f(x) on [a, b], ∫[a, b] f(x) dx = (1/2) * (f(a) + f(b)) * (b - a)

Question 32

What definite integral represents the area under the curve of the function f(x) = sin(x) between x = 0 and x = π?

Accepted Answer

∫[0,π] sin(x) dx

Answer

∫[0,π/2] sin(x) dx

Answer

∫[0,π] cos(x) dx

Answer

∫[0,2π] sin(x) dx

Question 33

What is the indefinite integral of the function f(x) = 1/x?

Accepted Answer

ln|x| + C

Answer

x + C

Answer

x² + C

Answer

1/x² + C

Question 34

Which integration technique is most suitable for evaluating the integral ∫x²e^x dx?

Accepted Answer

Integration by Parts

Answer

Substitution

Answer

Direct Integration

Answer

Partial Fractions

Question 35

If the derivative of a function F(x) is f(x), what is F(x) called?

Accepted Answer

Antiderivative of f(x)

Answer

Integral of f(x)

Answer

Derivative of f(x)

Answer

Differential of f(x)

Question 36

What is the value of the constant of integration 'C' in the indefinite integral ∫x dx = (x²/2) + C?

Accepted Answer

It can be any real number

Answer

1

Answer

0

Answer

It is not defined

Question 37

Which of these is NOT a property of definite integrals?

Accepted Answer

∫[a,b] f(x) dx = ∫[b,a] f(x) dx

Answer

∫[a,b] kf(x) dx = k∫[a,b] f(x) dx

Answer

∫[a,b] (f(x) + g(x)) dx = ∫[a,b] f(x) dx + ∫[a,b] g(x) dx

Answer

∫[a,b] f(x) dx = ∫[a,c] f(x) dx + ∫[c,b] f(x) dx

Question 38

What is the indefinite integral of the function  3x²?

Accepted Answer

x³ + C

Answer

x² + C

Answer

6x + C

Answer

3x³ + C

Question 39

Calculate the definite integral of f(x) = 2x from x = 1 to x = 3.

Accepted Answer

8

Answer

12

Answer

6

Answer

4

Question 40

What is the derivative of the function F(x) = ∫(x² + 1)dx from 0 to x?

Accepted Answer

x² + 1

Answer

x² + x

Answer

x²

Answer

2x

Question 41

Evaluate the definite integral of ∫(x³ + 2x)dx from x = 1 to x = 2.

Accepted Answer

15/4

Answer

11/4

Answer

7/4

Answer

9/4

Question 42

What is the indefinite integral of 1/x with respect to x?

Accepted Answer

ln|x| + C

Answer

x² + C

Answer

1/x² + C

Answer

x + C

Question 43

Which of the following represents the area under the curve of y = sin(x) from x = 0 to x = π?

Accepted Answer

∫(sin(x))dx from 0 to π

Answer

∫(sin(x))dx from 0 to 2π

Answer

∫(cos(x))dx from 0 to 2π

Answer

∫(cos(x))dx from 0 to π

Question 44

The integral of a function f(x) from a to b can be interpreted as the _____ enclosed by the curve of f(x), the x-axis, and the vertical lines x = a and x = b.

Accepted Answer

Area

Answer

Derivative

Answer

Slope

Question 45

What is the indefinite integral of e^x?

Accepted Answer

e^x + C

Answer

xe^x + C

Answer

1/x + C

Question 46

The integral of sin(x) is:

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Question 47

Find the indefinite integral of ∫(2x + 1) / (x² + x) dx.

Accepted Answer

ln|x² + x| + C

Answer

2x² + x + C

Answer

ln|x| + C

Answer

2ln|x² + x| + C