Integrals: Master Calculus with Interactive Quizzes

Question 1

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

Accepted Answer

1

Answer

-1

Answer

0

Answer

π/2

Question 2

Which of the following is NOT a method for finding integrals?

Accepted Answer

Chain Rule

Answer

Power Rule

Answer

Substitution Rule

Answer

Integration by Parts

Question 3

Which of the following improper integrals converges?

Accepted Answer

∫[0, ∞] e^(-x) dx

Answer

∫[1, ∞] 1/x^2 dx

Answer

∫[0, ∞] sin(x) dx

Answer

∫[1, ∞] ln(x) dx

Question 4

Calculate the indefinite integral of f(x) = x^2 + 2x - 1.

Accepted Answer

x^3 + x^2 - x + C

Answer

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

Answer

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

Answer

x^3 - x^2 + x + C

Question 5

Which of the following statements about the integral ∫(1/x) dx is true?

Accepted Answer

It is equal to ln(|x|) + C.

Answer

It is an improper integral.

Answer

It is a divergent integral.

Answer

It is equal to (1/2)x^2 + C.

Question 6

Find the area bounded by the curve y = x^2 + 1 and the x-axis over the interval [0, 2].

Accepted Answer

8/3

Answer

4/3

Answer

5

Answer

10

Question 7

Which of the following is NOT a technique used in the integration process?

Accepted Answer

Partial fractions

Answer

Power rule

Answer

Substitution

Answer

Trigonometric substitution

Question 8

Calculate the integral of f(x) = 1 / (x + 1)^2 dx.

Accepted Answer

-1 / (x + 1) + C

Answer

1 / (x + 1) + C

Answer

-1 / x + C

Answer

1 / x + C

Question 9

Which integral represents the area under the curve of y = 4x from x = 0 to x = 2?

Accepted Answer

∫ (4x) dx from 0 to 2

Answer

∫ (4x) dx from 2 to 0

Answer

∫ (2x) dx from 0 to 2

Answer

∫ (2x) dx from 2 to 0

Question 10

What is the integral of ∫ e^x dx?

Accepted Answer

e^x + C

Answer

2 e^x + C

Answer

1/2 e^x + C

Answer

x e^x + C

Question 11

Which of the following expresses the substitution rule for integrals?

Accepted Answer

∫ f(u(x)) u'(x) dx = F(u(x)) + C

Answer

∫ f(u(x)) u'(x) dx = f(u(x)) + C

Answer

∫ u'(x) f(x) dx = F(u(x)) + C

Answer

∫ u(x) f(x) dx = F(x) + C

Question 12

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

Accepted Answer

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

Answer

ln(x^2 + 1) + C

Answer

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

Answer

ln(x) + C

Question 13

Which of the following represents the integral of ∫ sin(2x) dx?

Accepted Answer

-cos(2x) + C

Answer

cos(2x) + C

Answer

1/2 cos(2x) + C

Answer

2 sin(2x) + C

Question 14

Which of the following is not a technique for finding integrals?

Accepted Answer

Integration by substitution

Answer

Power rule

Answer

Substitution

Answer

Integration by parts

Question 15

Find the integral of ∫x^2 dx:

Accepted Answer

x^3 / 3 + C

Answer

x^2 + C

Answer

x^3 + C

Answer

2x + C

Question 16

Which of the following is the correct expression for the integral ∫e^x dx?

Accepted Answer

e^x + C

Answer

e^x - C

Answer

x * e^x + C

Answer

x * e^x - C

Question 17

Find the integral of ∫(x + 2)^3 dx:

Accepted Answer

(x + 2)^4 / 4 + C

Answer

(x + 2)^3 + C

Answer

(x + 2)^2 + C

Answer

(x + 2)^4 + C

Question 18

Find the integral of ∫(x * ln(x)) dx:

Accepted Answer

x^2 * ln(x) - x^2 + C

Answer

x^2 * ln(x) + x^2 + C

Answer

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

Answer

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

Question 19

Calculate ∫e^x dx.

Accepted Answer

e^x + C

Answer

x*e^x + C

Answer

x^2*e^x + C

Answer

1/e^x + C

Question 20

Which substitution is correct for the integral ∫(x^2 + 1)^3 dx?

Accepted Answer

u = x^2 + 1

Answer

u = x

Answer

u = x^3

Answer

u = 1/x

Question 21

Evaluate the improper integral ∫(1/x) dx from 1 to infinity.

Accepted Answer

Diverges to infinity

Answer

Converges to 0

Answer

Converges to 1

Answer

Diverges to negative infinity

Question 22

Calculate the area under the curve y = x^2 from x = 0 to x = 2 using definite integrals.

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 23

Which technique is NOT used to find integrals?

Accepted Answer

Differentiation

Answer

Power Rule

Answer

Substitution

Answer

Integration by Parts

Question 24

Which of the following is NOT a technique used to find integrals?

Accepted Answer

Chain rule

Answer

Power rule

Answer

Substitution

Answer

Integration by parts

Question 25

Find the integral of ∫(x^2 + 2x) dx.

Accepted Answer

x^3/3 + x^2 + C

Answer

x^3 - x^2 + C

Answer

2x^3 + x^2 + C

Answer

x^2 + 2x + C

Question 26

Which of the following is the power rule for integrals?

Accepted Answer

∫x^n dx = (x^(n+1))/(n+1) + C

Answer

∫x^n dx = (n+1)x^(n-1) + C

Answer

∫x^n dx = n^x + C

Answer

∫x^n dx = 1/(n+1)x^(n-1) + C

Question 27

Which of the following integrals represents the area under the curve y = 2x from x = 0 to x = 3?

Accepted Answer

∫[0, 3] 2x dx

Answer

∫[0, 3] x^2 dx

Answer

∫[0, 3] 1/(2x) dx

Answer

∫[0, 3] 2(x^2 + 1) dx

Question 28

Find the integral of ∫(x^3 + 2x^2 - 5) dx.

Accepted Answer

(x^4)/4 + (2x^3)/3 - 5x + C

Answer

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

Answer

(x^4)/4 + (2x^3)/3 + 5x + C

Answer

(x^4)/4 - 2x^3 - 5x + C

Question 29

Which of the following is an indefinite integral of ∫sin(x) cos(x) dx?

Accepted Answer

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

Answer

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

Answer

sin(x) cos(x) + C

Answer

-sin(x) cos(x) + C

Question 30

Use the method of substitution to find the integral of the function f(x) = √x with respect to x.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 31

Determine the area of the region bounded by the curve y = x^2 and the x-axis from x = 0 to x = 2 using integrals.

Accepted Answer

8/3

Answer

4/3

Answer

16/3

Answer

32/3

Question 32

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 from x = 0 to x = 2.

Accepted Answer

64π/5

Answer

32π/5

Answer

128π/5

Answer

256π/5

Question 33

Evaluate the integral ∫e^x dx.

Accepted Answer

e^x + C

Answer

x * e^x + C

Answer

e^(x+1) + C

Answer

x * e^(x+1) + C

Question 34

Determine the area under the curve y = x^2 between x = 0 and x = 2.

Accepted Answer

8/3

Answer

4/3

Answer

16/3

Answer

32/3

Question 35

Which of the following is a definite integral?

Accepted Answer

∫0^2 (x^3 + 1) dx

Answer

∫x^2 dx

Answer

∫(x^2 + 2x) dx

Answer

∫e^x dx

Question 36

Find the integral of x^3 with respect to x.

Accepted Answer

(1 / 3) * x^4 + C

Answer

(1 / 4) * x^4 + C

Answer

x^4 + C

Answer

4x^3 + C

Question 37

Which substitution would be appropriate for evaluating the integral of √(1 - x^2) dx?

Accepted Answer

u = √(1 - x^2)

Answer

u = 1 - x^2

Answer

u = x^2

Answer

u = 1 / √(1 - x^2)

Question 38

Find the integral of ∫cos(x) dx.

Accepted Answer

sin(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Answer

-cos(x) + C

Question 39

Which technique should be used to evaluate the integral of ∫x * sin(x) dx?

Accepted Answer

Integration by parts

Answer

Substitution

Answer

Power rule

Answer

Trigonometric substitution

Question 40

Which of the following is the power rule of integration for indefinite integrals?

Accepted Answer

∫x^n dx = (x^(n+1))/(n+1) + C

Answer

∫e^x dx = e^x + C

Answer

∫sin(x) dx = -cos(x) + C

Answer

∫cos(x) dx = sin(x) + C

Question 41

Find the indefinite integral of ∫(2x + 3) dx:

Accepted Answer

2x^2 + 3x + C

Answer

x^2 + 3x + C

Answer

x^2 + 3 + C

Answer

2x^2 + 3 + C

Question 42

Which of the following is the power rule formula for integrals?

Accepted Answer

∫ x^n dx = x^(n+1) + C

Answer

∫ e^x dx = e^(x+1) + C

Answer

∫ sin(x) dx = -cos(x) + C

Answer

∫ ln(x) dx = x * ln(x) + C

Question 43

Which of the following is the integration by parts formula?

Accepted Answer

∫ u dv = uv - ∫ v du

Answer

∫ v du = uv - ∫ u dv

Answer

∫ udv = uv + ∫ v du

Answer

∫ vdu = uv + ∫ u dv

Question 44

Which of the following is a definite integral?

Accepted Answer

∫_a^b f(x) dx

Answer

∫ f(x) dx

Answer

f'(x) dx

Answer

∫ f(x) * g(x) dx

Question 45

Which of the following is the Fundamental Theorem of Calculus?

Accepted Answer

If f(x) is continuous on [a, b], then ∫_a^b f(x) dx = F(b) - F(a), where F(x) is any antiderivative of f(x)

Answer

If f(x) is continuous on [a, b], then ∫_a^b f(x) dx = F(b) + F(a), where F(x) is any antiderivative of f(x)

Answer

If f(x) is differentiable on [a, b], then ∫_a^b f(x) dx = f(b) - f(a)

Answer

If f(x) is differentiable on [a, b], then ∫_a^b f(x) dx = f(b) + f(a)

Question 46

Find the integral of the function: ∫sin(x) dx

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Answer

tan(x) + C

Question 47

Use the method of substitution to find the integral: ∫x * √(x² + 1) dx

Accepted Answer

(x²/2) * √(x² + 1) + C

Answer

(x²) * √(x² + 1) + C

Answer

(x³) * √(x² + 1) + C

Answer

Cannot be solved using substitution

Question 48

Which of the following is NOT a technique for finding integrals?

Accepted Answer

Chain Rule

Answer

Power Rule

Answer

Substitution Rule

Answer

Integration by Parts

Question 49

Find the integral of the function: ∫(x + 1) / √x dx

Accepted Answer

2 * √x + 2 * x^(3/2) + C

Answer

√x + x^(3/2) + C

Answer

x^(3/2) + C

Answer

Cannot be solved using integration by parts