Integrals: Definition, Techniques, and Applications - Online Learning Quizzes

Question 1

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

Accepted Answer

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

Answer

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

Answer

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

Answer

(x^5 / 4) + 2x^4 - x + C

Question 2

Find the indefinite integral of x * e^x dx.

Accepted Answer

xe^x - e^x + C

Answer

xe^x + e^x + C

Answer

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

Answer

x^2e^x + C

Question 3

Calculate the definite integral ∫[0, 2] x^3 - 3x + 2 dx.

Accepted Answer

6

Answer

4

Answer

8

Answer

10

Question 4

Determine whether the improper integral ∫[0, ∞] e^(-x^2) dx converges or diverges.

Accepted Answer

Converges

Answer

Diverges

Question 5

What is the antiderivative of  e^x * sin(x)?

Accepted Answer

-e^x * cos(x) + C

Answer

e^x * cos(x) + C

Answer

e^x * sin(x) + C

Question 6

Find the indefinite integral of the function f(x) = x^3 + 2x:

Accepted Answer

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

Answer

x^3 + x^2 + C

Answer

x^4 + 2x^2 + C

Answer

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

Question 7

Identify the integral that represents the arc length of the curve y = x^3/3 from x = 0 to x = 2:

Accepted Answer

∫[0, 2] √(1 + (x^2)^2) dx

Answer

∫[0, 2] √(x^2 + 1) dx

Answer

∫[0, 2] x^3 dx

Question 8

Evaluate the improper integral ∫[1, ∞] (1/x^2) dx:

Accepted Answer

1

Answer

0

Answer

∞

Answer

-1

Question 9

Which integration technique is most suitable for the function ∫(e^(x^2)) dx?

Accepted Answer

u-substitution

Answer

partial fractions

Answer

integration by parts

Answer

trigonometric substitution

Question 10

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

Accepted Answer

½ sin²(x) + C

Answer

-½ cos²(x) + C

Answer

cos²(x) + C

Question 11

Evaluate the integral ∫e^(2x) dx.

Accepted Answer

½ e^(2x) + C

Answer

e^(2x) + C

Answer

2 e^(2x) + C

Answer

½ e^(x) + C

Question 12

Find the definite integral of ∫[1, 3] (x² + 1) dx.

Accepted Answer

14/3

Answer

10/3

Answer

8/3

Answer

12/3

Question 13

Which of the following is the correct formula for the integral of (1/x) dx?

Accepted Answer

ln|x| + C

Answer

1/x + C

Answer

-ln|x| + C

Answer

x + C

Question 14

Which of the following techniques can be used to integrate the function ∫√x dx?

Accepted Answer

u-substitution

Answer

partial fractions

Answer

trigonometric substitution

Answer

integration by parts

Question 15

Find the integral of ∫(1/(1+x²))dx.

Accepted Answer

∫(1/(1+x²))dx = arctan(x) + C

Answer

∫(1/(1+x²))dx = tan(x) + C

Answer

∫(1/(1+x²))dx = cot(x) + C

Answer

∫(1/(1+x²))dx = sec(x) + C

Question 16

Determine if the improper integral ∫[1, ∞] (1/x)dx converges or diverges.

Accepted Answer

∫[1, ∞] (1/x)dx diverges

Answer

∫[1, ∞] (1/x)dx converges to 0

Answer

∫[1, ∞] (1/x)dx converges to -1

Answer

∫[1, ∞] (1/x)dx converges to 1

Question 17

Select the application of integrals in statistics.

Accepted Answer

Calculating probability distributions

Answer

Computing work done by a force

Answer

Finding the volume of solids

Answer

Solving systems of equations

Question 18

Identify a property of the integral operator.

Accepted Answer

Linearity

Answer

Associativity

Answer

Commutativity

Answer

Invertibility

Question 19

Obtain the solution to the differential equation dy/dx = x + y using integration.

Accepted Answer

y = (x²)/2 + xy + C

Answer

y = (x²)/2 - xy + C

Answer

y = x - y + C

Answer

y = x + y + C

Question 20

Calculate the definite integral ∫[0, π/2] sin(x) dx.

Accepted Answer

1

Answer

π/2

Answer

-1

Answer

0

Question 21

Which of the following improper integrals is convergent?

Accepted Answer

∫[1, ∞] 1/x^2 dx

Answer

∫[0, 1] 1/x dx

Answer

∫[-1, 1] 1/(x^2 - 1) dx

Question 22

Find the indefinite integral of x² with respect to x.

Accepted Answer

x³/3 + C

Answer

x³ + C

Answer

2x + C

Answer

2x² + C

Question 23

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

Accepted Answer

∫u dv = uv - ∫v du

Answer

∫u dv = ∫v du

Answer

∫u dv = u + v - ∫v du

Answer

∫u dv = uv + ∫v du

Question 24

Find the indefinite integral of 1/x with respect to x.

Accepted Answer

ln|x| + C

Answer

1/x² + C

Answer

-1/x² + C

Answer

x + C

Question 25

The integral of a velocity function over a time interval represents:

Accepted Answer

Displacement

Answer

Momentum

Answer

Acceleration

Answer

Speed

Question 26

Which integration technique is suitable for a function involving a product of two functions?

Accepted Answer

Integration by Parts

Answer

Substitution

Answer

Partial Fractions

Answer

Direct Integration

Question 27

Find the indefinite integral of cos(x) with respect to x.

Accepted Answer

sin(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Answer

-cos(x) + C

Question 28

Which of the following is NOT a valid integration technique?

Accepted Answer

Quadratic Formula

Answer

Substitution

Answer

Integration by Parts

Answer

Partial Fractions

Question 29

Which of the following is NOT a fundamental theorem of calculus?

Accepted Answer

Integration by substitution

Answer

The derivative of an integral is the integrand

Answer

The definite integral of a continuous function over a closed interval is a finite number

Answer

The integral of a derivative is the original function

Question 30

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

Accepted Answer

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

Answer

(2x) + cos(x) + C

Answer

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

Answer

(x^3) - sin(x) + C

Question 31

Which of the following techniques is used to integrate the function ∫(x * e^x) dx?

Accepted Answer

Integration by parts

Answer

Partial fractions

Answer

Trigonometric substitution

Answer

Substitution

Question 32

What is the value of the definite integral ∫[0, π](cos(x)) dx?

Accepted Answer

0

Answer

π

Answer

1

Answer

2

Question 33

Which of the following integrals represents the area under the curve y = x^2 from x = 0 to x = 2?

Accepted Answer

∫[0, 2](x^2) dx

Answer

∫[2, 0](x^2) dx

Answer

∫[0, 2] 2x dx

Answer

∫[0, 2] x dx

Question 34

What is the order of the differential equation y''' + 2y'' - 5y' + 6y = 0?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 35

Which of the following is a solution to the differential equation dy/dx = xy?

Accepted Answer

y = Ce^(x^2/2)

Answer

y = C/x

Answer

y = Cx^2

Answer

y = Csin(x)

Question 36

What is the general solution to the differential equation y' + y = e^x?

Accepted Answer

y = Ce^(-x) + e^x

Answer

y = Ce^(-x) - e^x

Answer

y = Ce^x + e^(-x)

Answer

y = Ce^x - e^(-x)

Question 37

Which of the following is a technique used to solve the differential equation y'' + 9y = 0?

Accepted Answer

Characteristic equation

Answer

Integrating factor

Answer

Variation of parameters

Answer

Laplace transform

Question 38

What is the Laplace transform of the function f(t) = t * e^(-2t)?

Accepted Answer

1 / (s+2)^2

Answer

1 / s^2

Answer

s / (s+2)^2

Answer

(s+2) / s^2