Integrals: Master Calculus with Interactive Quizzes

Question 1

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

Accepted Answer

Integration by logarithmic differentiation

Answer

Integration by substitution

Answer

Integration by parts

Answer

Integration by trigonometric substitution

Question 2

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 3

Determine the integral of (1/√(1 - x^2)).

Accepted Answer

arcsin(x) + C

Answer

arccos(x) + C

Answer

sin(x) + C

Answer

cos(x) + C

Question 4

Which of the following is NOT a valid method for evaluating integrals?

Accepted Answer

Graphical estimation

Answer

Substitution

Answer

Integration by parts

Question 5

What is the correct formula for the indefinite integral of x^2?

Accepted Answer

x^3/3 + C

Answer

2x + C

Answer

x^3 + C

Answer

3x^2 + C

Question 6

Calculate the definite integral of 2x from 0 to 2.

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 7

Find the integral of sin(x).

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Question 8

What is the integral of e^x?

Accepted Answer

e^x + C

Answer

xe^x + C

Answer

e^x / x + C

Answer

x / e^x + C

Question 9

Which of the following is the integral of sin(x) with respect to x?

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(x)

Answer

-sin(x)

Question 10

Evaluate the improper integral ∫[0,∞] e^(-x) dx

Accepted Answer

1

Answer

0

Answer

2

Answer

∞

Question 11

Which of the following is a method for integrating rational functions?

Accepted Answer

Partial fractions

Answer

Long division

Answer

Integration by parts

Answer

Trigonometric identities

Question 12

Find the area under the curve y = x^3 between x = 0 and x = 2

Accepted Answer

8

Answer

2

Answer

4

Answer

16

Question 13

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

Accepted Answer

The Fundamental Theorem of Algebra

Answer

The Chain Rule

Answer

The Mean Value Theorem

Answer

The Second Derivative Test

Question 14

Which of the following is a substitution that can be used to integrate ∫(x^2 + 1)^3 x dx?

Accepted Answer

u = x^2 + 1

Answer

u = x

Answer

u = x^3

Answer

u = 1/x

Question 15

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

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Question 16

Which technique is commonly used to find the integral of a rational function?

Accepted Answer

Partial fractions

Answer

Integration by parts

Answer

Trigonometric substitution

Question 17

Calculate the definite integral of f(x) = x^2 from 0 to 2.

Accepted Answer

4

Answer

2

Question 18

Find the integral of sec^2(x) with respect to x.

Accepted Answer

tan(x) + C

Answer

sec(x) + C

Question 19

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

Accepted Answer

Graphical approximation

Answer

Substitution method

Answer

Integration by parts

Answer

Trigonometric substitution

Question 20

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

Accepted Answer

Factoring

Answer

Substitution

Answer

Integration by Parts

Answer

Partial Fractions

Question 21

What is the value of the integral ∫sin(x) dx from 0 to π/2?

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 22

Which technique should you use to integrate the function ∫√(x) dx?

Accepted Answer

U-substitution

Answer

Integration by parts

Answer

Partial fractions

Answer

Trigonometric substitution

Question 23

Use integration by parts to evaluate the integral ∫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^3/3)e^x + C

Question 24

Which of the following is an invalid technique for finding integrals?

Accepted Answer

Guessing

Answer

Substitution

Answer

Integration by parts

Answer

Partial fractions

Question 25

Find the area under the curve of the function f(x) = x^2 - 3x between x = 2 and x = 5.

Accepted Answer

20.5

Answer

0

Answer

41

Question 26

Determine whether the function f(x) = x^3 - 3x^2 + 2x + 1 is increasing or decreasing on the interval [-1, 2] using definite integrals.

Accepted Answer

Increasing

Answer

Decreasing

Answer

Concave up

Answer

Concave down

Question 27

Which function has a derivative of x² + 1?

Accepted Answer

x³/3 + x + C

Answer

x²/2 + x + C

Answer

x³ + 1

Answer

(x² + 1)dx

Question 28

Evaluate the integral of ∫(3x + 2) dx.

Accepted Answer

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

Answer

(3x^2) + 2x + C

Answer

(3x) + 2 + C

Answer

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

Question 29

Which of the following is the correct substitution to evaluate the integral of ∫cos(x^2) dx?

Accepted Answer

u = x^2

Answer

u = cos(x)

Answer

u = sin(x)

Answer

u = e^x

Question 30

What is the indefinite integral of ∫(1/x) dx?

Accepted Answer

ln(|x|) + C

Answer

1/x + C

Answer

x + C

Answer

x^2 + C

Question 31

Find the integral of the function f(x) = 1/(x - 1).

Accepted Answer

ln|x - 1| + C

Answer

1/x - 1 + C

Answer

x - 1 + C

Answer

-1/x - 1 + C

Question 32

Find the integral of the function f(x) = sin(x)cos(x).

Accepted Answer

1/2sin^2(x) + C

Answer

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

Answer

sin(x)cos(x) + C

Question 33

Find the integral of the function f(x) = 1/(x^2 + 1).

Accepted Answer

arctan(x) + C

Answer

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

Answer

ln(x^2 + 1) + C

Answer

1/(x^2 + 1) + C

Question 34

Which expression represents the integral of the function f(x) = 2x?

Accepted Answer

x² + C

Answer

x + C

Answer

2x²

Question 35

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

Accepted Answer

-cos(x) + C

Answer

sin(x) + C

Answer

cos(x) + C

Question 36

The area under the curve y = x³ between x = 0 and x = 2 is given by which definite integral?

Accepted Answer

∫[0, 2] x³ dx

Answer

∫[0, 2] 3x dx

Answer

∫[0, 2] x² dx

Answer

∫[0, 2] 3x² dx

Question 37

Which integration technique is most suitable for finding the integral of the function f(x) = x² sin(x)?

Accepted Answer

Integration by Parts

Answer

Direct Integration

Answer

Substitution

Answer

Partial Fractions

Question 38

The integral of the function f(x) = 1/x is:

Accepted Answer

ln|x| + C

Answer

-1/x² + C

Answer

x + C

Question 39

Which property of definite integrals is incorrect?

Accepted Answer

∫[a, b] f(x) dx = ∫[b, a] f(x) dx

Answer

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

Question 40

The average value of the function f(x) = x² over the interval [0, 2] is:

Accepted Answer

4/3

Answer

2

Answer

1

Answer

8/3

Question 41

What is the integral of the function f(x) = x^2 + 2x?

Accepted Answer

∫(x^2 + 2x) dx = (1/3)x^3 + x^2 + C

Answer

∫(x^2 + 2x) dx = (1/2)x^3 + x^2 + C

Answer

∫(x^2 + 2x) dx = x^3 + 2x^2 + C

Answer

∫(x^2 + 2x) dx = x^3 + C

Question 42

What is the indefinite integral of  (e^x + sin(x)) with respect to x?

Accepted Answer

e^x - cos(x) + C

Answer

e^x - 2cos(x) + C

Answer

2e^x - 2cos(x) + C

Answer

e^x + cos(x) + C

Question 43

Which expression represents the antiderivative of the function f(x) = cos(x)?

Accepted Answer

sin(x) + C

Answer

-sin(x) + C

Answer

tan(x) + C

Answer

sec(x) + C

Question 44

Which integration technique is most suitable for finding the integral of the function f(x) = (x^2 + 1)/(x - 1)?

Accepted Answer

Partial Fraction Decomposition

Answer

Trigonometric Substitution

Answer

U-Substitution

Answer

Integration by Parts

Question 45

What is the value of the improper integral ∫[1, ∞] (1/x^2) dx?

Accepted Answer

1

Answer

0

Answer

-1

Answer

∞

Question 46

What is the volume of the solid formed by rotating the area enclosed by the curves y = x^2 and y = 4 around the x-axis?

Accepted Answer

(32π/5)

Answer

(16π/5)

Answer

(64π/5)

Answer

(8π/5)

Question 47

Calculate the work done by a force represented by f(x) = x^2 + 2x over the interval [0, 2].

Accepted Answer

(14/3)

Answer

(7/3)

Answer

(28/3)

Answer

(8/3)

Question 48

What is the integral of x^2 with respect to x?

Accepted Answer

x^3/3 + C

Answer

2x^2 + C

Answer

2x + C

Answer

x^3 + C

Question 49

What is the integral of sin(x) with respect to x?

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C

Question 50

What is the area under the curve of y = x^2 from x = 0 to x = 2?

Accepted Answer

8/3

Answer

16/3

Answer

4

Answer

2

Question 51

What is the integral of e^x with respect to x?

Accepted Answer

e^x + C

Answer

e^(x+1) + C

Answer

x*e^x + C

Answer

ln(x) + C

Question 52

The integral of '1/x' with respect to x (for x > 0) is:

Accepted Answer

ln(x) + C

Answer

x + C

Answer

x^2/2 + C

Answer

-1/x^2 + C

Question 53

A particle moves along a straight line with velocity v(t) = 2t + 1. What is the displacement of the particle from t = 0 to t = 2?

Accepted Answer

6

Answer

4

Answer

2

Question 54

The integral of '1/(1+x^2)' with respect to x is:

Accepted Answer

arctan(x) + C

Answer

ln(1+x^2) + C

Answer

x/(1+x^2) + C

Answer

1/(2x) + C