Master Integral Calculus: Online Quizzes and Practice Questions

Question 1

Calculate the area under the curve y = x^2 + 1 between x = 0 and x = 2 using the definite integral:

Accepted Answer

6

Answer

4

Answer

8

Answer

10

Question 2

Which integration technique is most appropriate for evaluating the integral ∫(x^2 - 1)/(x + 1) dx?

Accepted Answer

partial fractions

Answer

u-substitution

Answer

integration by parts

Answer

trigonometric substitution

Question 3

Calculate the integral ∫e^(3x) cos(2x) dx:

Accepted Answer

(1/2)e^(3x) sin(2x) + (1/6)e^(3x) cos(2x) + C

Answer

(1/2)e^(3x) cos(2x) - (1/6)e^(3x) sin(2x) + C

Answer

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

Answer

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

Question 4

Which of the following formulas represents the volume of a cone with radius r and height h?

Accepted Answer

(1/3)πr^2h

Answer

(1/2)πr^2h

Answer

πr^2h

Answer

2πr^2h

Question 5

Which formula represents the integral of 1/(x² + 1)?

Accepted Answer

arctan(x) + C

Answer

tan(x) + C

Answer

sec(x) + C

Answer

1/(x² + 1) + C

Question 6

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 7

Evaluate the integral ∫₀^(π/2) sin(x) dx

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 8

Find the area under the curve y = x² between x = 0 and x = 2.

Accepted Answer

4

Answer

1

Answer

2

Answer

8

Question 9

Which is NOT a property of integrals?

Accepted Answer

The integral of a function with respect to a variable is independent of the choice of variable.

Answer

The integral of a constant is equal to the constant multiplied by the variable of integration.

Answer

The integral of a sum of functions is equal to the sum of the integrals of the individual functions.

Answer

The integral of a product of functions is equal to the product of the integrals of the individual functions.

Question 10

Find the centroid of the region bounded by y = √x and y = 2.

Accepted Answer

(4/5, 4/3)

Answer

(5/4, 3/4)

Answer

(3/4, 4/5)

Answer

(4/3, 5/4)

Question 11

Which of the following definitions describes an integral?

Accepted Answer

The area under a curve over a specified interval

Answer

The slope of a curve at a given point

Answer

The volume of a solid generated by rotating a region

Answer

The derivative of a function

Question 12

The Fundamental Theorem of Calculus establishes a crucial connection between what two mathematical concepts?

Accepted Answer

Integrals and derivatives

Answer

Integrals and limits

Answer

Derivatives and limits

Answer

Integrals and volumes

Question 13

Which of the listed options is NOT a recognized technique used to evaluate integrals?

Accepted Answer

Differentiation

Answer

Integration by substitution

Answer

Integration by parts

Answer

Integration by trigonometric identities

Question 14

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

Accepted Answer

(x^3)/3 + C

Answer

x^2 + C

Answer

x + C

Answer

2x + C

Question 15

To calculate the area under the curve y = sin(x) between x = 0 and x = π, which integral should be utilized?

Accepted Answer

∫[0,π] sin(x) dx

Answer

∫[π,0] sin(x) dx

Answer

π - ∫[0,π] sin(x) dx

Answer

π + ∫[0,π] sin(x) dx

Question 16

Which property of integrals is employed to evaluate the integral of a sum of functions?

Accepted Answer

Linearity

Answer

Product rule

Answer

Substitution rule

Answer

Integration by parts

Question 17

If f(x) is a continuous function on the interval [a, b], what does the integral of f(x) from a to b represent?

Accepted Answer

The area under the curve y = f(x) from a to b

Answer

The net area between the curve y = f(x) and the x-axis from a to b

Answer

The volume of the solid generated by rotating the region bounded by the curve y = f(x) and the x-axis from a to b about the x-axis

Answer

The length of the curve y = f(x) from a to b

Question 18

Evaluate the definite integral of ∫ from (0 to 2) (x^2 + sin(x)) dx.

Accepted Answer

14/3

Answer

10/3

Answer

8/3

Answer

12/3

Question 19

Use the method of substitution to find the integral of ∫(x^2 + 1)^3 * 2x dx.

Accepted Answer

(x^2 + 1)^4/2 + C

Answer

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

Answer

(x^2 + 1)^3 + C

Answer

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

Question 20

Which of the following is the correct integral for finding the volume of the solid generated by rotating the region bounded by y = x^2 and y = 4 about the x-axis?

Accepted Answer

∫ from (0 to 2) π(4 - x^2)^2 dx

Answer

∫ from (0 to 2) π(x^2 - 4)^2 dx

Answer

∫ from (0 to 2) π(4 - x^2) dx

Answer

∫ from (0 to 2) π(x^2 - 4) dx

Question 21

Evaluate the indefinite integral ∫(x^2 + 2x - 1) dx:

Accepted Answer

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

Answer

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

Answer

x^3 + x^2 - 1 + C

Answer

2x^3 + x^2 - 1 + C

Question 22

Find the integral of ∫(x * e^x) dx.

Accepted Answer

x * e^x - e^x + C

Answer

(x - 1) * e^x + C

Answer

(x + 1) * e^x - e^x + C

Answer

x * e^x + C

Question 23

Which technique is most suitable for evaluating the integral ∫(x^2 + 1) / (x - 1) dx?

Accepted Answer

Partial fractions

Answer

Substitution

Answer

Integration by parts

Answer

Trigonometric substitution

Question 24

Determine the area under the curve y = sin(x) between the x-values 0 and π/2.

Accepted Answer

1

Answer

0

Answer

2

Answer

π

Question 25

State the chain rule for derivatives.

Accepted Answer

d/dx(f(g(x))) = f'(g(x)) * g'(x)

Answer

d/dx(f(g(x))) = g'(x) * f'(g(x))

Answer

d/dx(f(g(x))) = (f(g(x)))' * (g(x))'

Answer

d/dx(f(g(x))) = (g(f(x)))' * (f(x))'

Question 26

Find the derivative of the function f(x) = (x^2 + 1) * sin(x).

Accepted Answer

2x * sin(x) + (x^2 + 1) * cos(x)

Answer

(x^2 + 1) * sin(x) - 2x * cos(x)

Answer

2x * cos(x) + (x^2 + 1) * cos(x)

Answer

2x * sin(x) - (x^2 + 1) * cos(x)

Question 27

Determine the limit of the function f(x) = (x^3 - 2x^2 + 1) / (x^2 - 1) as x approaches infinity.

Accepted Answer

1

Answer

∞

Answer

-∞

Answer

0

Question 28

Write out the Taylor series expansion of f(x) = e^x around the point x = 0.

Accepted Answer

1 + x + (x^2) / 2! + (x^3) / 3! + ...

Answer

x + (x^2) / 2! + (x^3) / 3! + (x^4) / 4! + ...

Answer

1 + x^2 + (x^3) / 2! + (x^4) / 3! + ...

Answer

x + x^2 + x^3 + x^4 + ...

Question 29

Compute the integral ∫(x^2 + 3x) dx:

Accepted Answer

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

Answer

x^3/3 + 3x + C

Answer

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

Answer

2x^3/3 + 3x + C

Question 30

Which integral represents the area bounded by the curve y = f(x), the x-axis, and vertical lines x = a and x = b?

Accepted Answer

∫[a, b] f(x) dx

Answer

∫[b, a] f(x) dx

Answer

∫[a, b] x f(x) dx

Answer

∫[a, b] f'(x) dx

Question 31

Which technique is used to evaluate integrals of the form ∫f(x)g'(x) dx?

Accepted Answer

Integration by parts

Answer

Integration by substitution

Answer

Integration by partial fractions

Answer

Integration by series expansion

Question 32

Which integral represents the volume generated by rotating the region bounded by y = x^2 and y = 4 about the x-axis?

Accepted Answer

π∫[0, 2] (4 - x^2)^2 dx

Answer

π∫[0, 2] (x^2 - 4)^2 dx

Answer

∫[0, 2] (4 - x^2)^2 dx

Answer

∫[0, 2] (x^2 - 4)^2 dx

Question 33

Which condition is necessary for the improper integral ∫[a, ∞) f(x) dx to converge?

Accepted Answer

f(x) approaches zero as x approaches infinity

Answer

f(x) approaches a finite value as x approaches infinity

Answer

f(x) is continuous on [a, ∞)

Answer

f(x) is differentiable on [a, ∞)

Question 34

Which integral is used to calculate the arc length of the curve y = f(x) from a to b?

Accepted Answer

∫[a, b] √(1 + (dy/dx)^2) dx

Answer

∫[a, b] √(1 - (dy/dx)^2) dx

Answer

∫[a, b] (dy/dx) dx

Answer

∫[a, b] f(x) dx

Question 35

Which integration technique is used to find the integral ∫(x * e^x) dx?

Accepted Answer

Integration by parts

Answer

Substitution

Answer

Partial fractions

Answer

Trig substitution

Question 36

Which of the following is NOT a type of improper integral?

Accepted Answer

Definite integral

Answer

Type I

Answer

Type II

Answer

Type III

Question 37

Find the derivative of the function ∫(e^t) dt

Accepted Answer

e^t

Answer

t * e^t

Answer

e^(t^2)

Answer

t * e^(t^2)

Question 38

Which method is used to find the volume of a solid generated by rotating a curve around the x-axis?

Accepted Answer

Disk method

Answer

Shell method

Answer

Cylindrical shells method

Answer

Washer method

Question 39

Find the area under the curve y = e^x from x = 0 to x = 1.

Accepted Answer

e

Answer

e-1

Answer

e+1

Answer

2e

Question 40

Find the definite integral of f(x) = |x-2| over the interval [1, 5].

Accepted Answer

8

Answer

6

Answer

10

Answer

12

Question 41

Which property of definite integrals is NOT valid?

Accepted Answer

Additivity under subtraction

Answer

Reversal of limits

Answer

Linearity

Answer

Fundamental Theorem of Calculus

Question 42

Integrate the function: ∫cos(x) dx

Accepted Answer

sin(x) + C

Answer

tan(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C

Question 43

Determine the integral that represents the length of the curve y = x³ from x = 0 to x = 1.

Accepted Answer

∫√(1 + (dy/dx)²) dx

Answer

∫√(1 + x²) dx

Answer

∫√(1 + y²) dx

Answer

∫√(y² - 1) dx

Question 44

What is the integral of 3x² dx?

Accepted Answer

x³ + C

Answer

9x + C

Answer

6x + C

Question 45

Find the area bounded by the curve y = x² + 1, the x-axis, and the lines x = 0 and x = 2.

Accepted Answer

14/3

Answer

10/3

Answer

8/3

Answer

4

Question 46

Evaluate the definite integral ∫[1, 3] (2x - 1) dx.

Accepted Answer

6

Answer

8

Answer

2

Answer

4

Question 47

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

Accepted Answer

Substitution Method

Answer

Integration by Parts

Answer

Trigonometric Substitution

Answer

Partial Fractions

Question 48

The volume of the solid generated by rotating the region bounded by the curve y = √x, the x-axis, and the line x = 4 about the x-axis is given by:

Accepted Answer

π∫[0, 4] (√x)² dx

Answer

2π∫[0, 4] x√x dx

Answer

π∫[0, 4] x² dx

Answer

2π∫[0, 4] √x dx

Question 49

Find the integral of sin(2x) dx.

Accepted Answer

-1/2 cos(2x) + C

Answer

2 cos(2x) + C

Answer

1/2 cos(2x) + C

Answer

-2 cos(2x) + C

Question 50

What is the derivative of the integral ∫[1, x] (t² + 1) dt?

Accepted Answer

x² + 1

Answer

2x + 1

Answer

t² + 1

Question 51

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

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 52

The integral ∫(1/x) dx represents:

Accepted Answer

The natural logarithm of the absolute value of x

Answer

The square of x

Answer

The reciprocal of x

Question 53

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

Accepted Answer

∫u dv = uv - ∫v du

Answer

∫u dv = ∫v du - ∫u du

Answer

∫u dv = uv + ∫v du

Answer

∫u dv = ∫u du - ∫v dv