Master Integral Calculus: Practice Quizzes and Learning Resources

Question 1

Evaluate the integral of sin(x) cos(x) over the interval [0, π/2].

Accepted Answer

1/2

Answer

-1/2

Answer

0

Answer

1

Question 2

Find the area between the curves y = x^2 and y = 4.

Accepted Answer

5.33

Answer

2.67

Answer

16

Answer

32

Question 3

Which of the following is NOT a property of definite integrals?

Accepted Answer

∫[a, b] f(x) dx = 0 if f(x) is an odd function

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

Question 4

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

Accepted Answer

∫u dv = uv - ∫v du

Answer

∫v du = uv - ∫u dv

Answer

∫u dv = u dv - ∫v du

Answer

∫v du = u dv - ∫u dv

Question 5

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

Accepted Answer

cos(x) + C

Answer

-cos(x) + C

Answer

sin(x) + C

Question 6

Which technique is used to integrate rational functions?

Accepted Answer

Partial fractions

Answer

Trig substitution

Question 7

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

Accepted Answer

cos(x) + C

Answer

-cos(x) + C

Answer

cos(x) - C

Answer

-cos(x) - C

Question 8

Find the definite integral of ∫[0,π] sin(x) dx.

Accepted Answer

1

Answer

2

Answer

-1

Answer

0

Question 9

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

Accepted Answer

converges but not to 1

Answer

1

Answer

diverges

Answer

cannot be determined

Question 10

If ∫f(x) dx = x^3 + 2x + C, what is the derivative of f(x)?

Accepted Answer

3x^2 + 2

Answer

x^3 + 2

Answer

x^2 + 2

Answer

The derivative cannot be determined from the given information.

Question 11

What is the result of integrating the function f(x) = e^(2x) with respect to x?

Accepted Answer

1/2 e^(2x) + C

Answer

2 e^(2x) + C

Answer

1/2 e^(2x) - C

Answer

2 e^(2x) - C

Question 12

Which of the following techniques can be used to integrate ∫(x^2 + 1)/(x + 1) dx?

Accepted Answer

Integration by partial fractions

Answer

Integration by substitution

Answer

Integration by trigonometric substitution

Answer

Integration by integration by parts

Question 13

Evaluate ∫(2x^3 + sin(x)) dx.

Accepted Answer

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

Answer

x^4 - cos(x) + C

Answer

x^4 + sin(x) + C

Answer

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

Question 14

Which integral represents the area under the curve y = x^2 + 1 from x = 0 to x = 2?

Accepted Answer

∫(x^2 + 1) dx from 0 to 2

Answer

∫(2x + 1) dx from 0 to 2

Answer

∫x^2 dx from 0 to 2 + ∫1 dx from 0 to 2

Answer

∫(x + 1)^2 dx from 0 to 2

Question 15

Find the volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 4 from x = 0 to x = 2 about the x-axis.

Accepted Answer

16π/5 cubic units

Answer

8π/5 cubic units

Answer

32π/5 cubic units

Answer

64π/5 cubic units

Question 16

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

Accepted Answer

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

Answer

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

Answer

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

Answer

(x^2/3) + 2x + 1 + C

Question 17

Calculate the definite integral of sin(x) from x = 0 to x = π/2.

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 18

Find the area of the region bounded by the curves y = x^2 and y = 4.

Accepted Answer

16/3

Answer

8/3

Answer

32/3

Answer

64/3

Question 19

Find the Laplace transform of e^(-at).

Accepted Answer

1/(s + a)

Answer

1/(s - a)

Answer

a/(s + a)

Answer

a/(s - a)

Question 20

Use the method of substitution to find the integral of ∫(cos²(x))sin(x) dx.

Accepted Answer

-cos³(x) + C

Answer

cos³(x) + C

Answer

-1/3cos³(x) + C

Answer

1/3cos³(x) + C

Question 21

Calculate the indefinite integral of (x^2 + 3x - 5) dx.

Accepted Answer

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

Answer

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

Answer

3x^2 + 3x + C

Answer

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

Question 22

Which statement about the properties of definite integrals is incorrect?

Accepted Answer

The definite integral of a continuous function is always continuous.

Answer

The result of a definite integral is numerical.

Answer

The value of a definite integral can be negative.

Answer

The definite integral of a function over an interval equals the area under the curve.

Question 23

Find the integral of (sin(x) + cos(x)) dx.

Accepted Answer

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

Answer

cos(x) - sin(x) + C

Answer

sin(x) - cos(x) + C

Answer

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

Question 24

What is the correct integral form of e^x dx?

Accepted Answer

e^x + C

Answer

e^x/x + C

Answer

xe^x + C

Answer

x/e^x + C

Question 25

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

Accepted Answer

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

Answer

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

Answer

1/2 * arctan(x - 1) + 1/2 * arctan(x + 1) + C

Answer

arctan(x - 1) + arctan(x + 1) + C

Question 26

Calculate the indefinite integral of x^2 dx.

Accepted Answer

x^3/3 + C

Answer

x^3 + C

Answer

3x^2 + C

Answer

x^2 + C

Question 27

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

Accepted Answer

1

Answer

0

Answer

-1

Answer

π/2

Question 28

Which integration technique is appropriate for evaluating ∫(x + 1)^2 dx?

Accepted Answer

Substitution

Answer

Integration by parts

Answer

Partial fractions

Answer

Trigonometric substitution

Question 29

Calculate the area bounded by the curve y = x^3 between x = 0 and x = 2.

Accepted Answer

16/5

Answer

8/5

Answer

4

Answer

2

Question 30

Find the integral of 1/(x-1) dx.

Accepted Answer

ln(|x-1|) + C

Answer

1/(x-1) + C

Answer

-ln(|x-1|) + C

Answer

-1/(x-1) + C

Question 31

Find the integral of e^x dx.

Accepted Answer

e^x + C

Answer

xe^x + C

Answer

x + C

Answer

1/e^x + C

Question 32

Find the integral of ∫(x^2 + 1)/(x-1) dx using partial fractions.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 33

Evaluate the indefinite integral ∫(x² + 2x - 1) dx.

Accepted Answer

x³/3 + x² - x + C

Answer

x³ + 2x² - x + C

Answer

(x² + 2x - 1)² + C

Answer

Cannot be integrated

Question 34

Which of the following techniques can be used to integrate ∫cosx dx?

Accepted Answer

Both integration by parts and trigonometric identities

Answer

Integration by parts

Answer

Substitution

Answer

Trigonometric identities

Question 35

Which of the following is a solution to the differential equation dy/dx = x² + 2x - 3?

Accepted Answer

y = x³/3 + x² - 3x + C

Answer

y = x⁴/4 + x³ - 3x² + C

Answer

y = x²/2 + x - 3x + C

Answer

y = x³/3 + x² - 3 + C

Question 36

Evaluate the indefinite integral of ∫e^x dx:

Accepted Answer

ex + C

Answer

ex - C

Answer

x*ex + C

Answer

x/ex + C

Question 37

Which of the following is the result of finding the indefinite integral of 1/(x^2 + 1) with respect to x?

Accepted Answer

tan^(-1)x + C

Answer

cot^(-1)x + C

Answer

sin^(-1)x + C

Answer

cos^(-1)x + C

Question 38

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

2∫0^2 x^2 dx

Answer

∫0^2 2x dx

Question 39

Which of the following is a valid substitution to find the integral of ∫x*cos(x^2) dx?

Accepted Answer

u = x^2

Answer

u = cos(x^2)

Answer

u = sin(x^2)

Answer

u = x

Question 40

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

Accepted Answer

∫(x^2) dx from 0 to 2

Answer

∫(x^2) dx from 2 to 0

Answer

∫(2x) dx from 0 to 2

Answer

∫(2x) dx from 2 to 0

Question 41

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

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 42

For the integral ∫(√(x^2 + 1)) dx, the most appropriate substitution is:

Accepted Answer

u = √(x^2 + 1)

Answer

u = x^2 + 1

Answer

u = x

Answer

u = 1 / (x^2 + 1)

Question 43

The integral of ∫(e^x + ln(x)) dx is:

Accepted Answer

e^x + x * ln(x)

Answer

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

Answer

x * e^x + C

Answer

None of the choices are correct

Question 44

The integral of ∫(1 / (x + 1)) dx is:

Accepted Answer

ln(|x + 1|) + C

Answer

ln(x + 1) + C

Answer

1 / (x + 1) + C

Answer

None of the choices are correct

Question 45

The integral of ∫(cos(x) * sin(x)) dx is:

Accepted Answer

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

Answer

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

Answer

sin(x) * cos(x) + C

Answer

None of the choices are correct

Question 46

The integral of f(x) = x^2 is:

Accepted Answer

x^3/3 + C

Answer

x^3 + C

Answer

2x + C

Answer

x^2/2 + C

Question 47

To integrate (x^2 + 1)/(x+1), the most appropriate technique is:

Accepted Answer

Partial fractions

Answer

Substitution

Answer

Integration by parts

Answer

Trigonometric substitution

Question 48

The antiderivative of f(x) = cos(x) + e^x is:

Accepted Answer

sin(x) + e^x + C

Answer

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

Answer

sin(x) - e^x + C

Answer

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

Question 49

Indicate the integral that measures the area under the curve of y = x³ between the boundaries x = 0 and x = 2.

Accepted Answer

∫[0,2] x³ dx

Answer

∫[2,0] x³ dx

Answer

∫[0,2] 2x dx

Answer

∫[2,0] 2x dx

Question 50

What is the correct formula for calculating the work done by a force F(x) acting over a distance x?

Accepted Answer

∫[a,b] F(x) dx

Answer

∫[b,a] F(x) dx

Answer

F(x) ∫[a,b] dx

Answer

F(x) ∫[b,a] dx

Question 51

Which of the following is NOT an advantage of using integration?

Accepted Answer

Simplifying algebraic expressions

Answer

Finding areas and volumes

Answer

Calculating work done by forces

Answer

Solving differential equations

Question 52

Write down the integral that represents the area under the curve y = 2x² - 5 between the limits x = 1 and x = 3.

Accepted Answer

∫(2x² - 5) dx from x = 1 to x = 3

Answer

∫(2x² - 5) dx

Answer

∫(2x²) dx - 5∫dx

Answer

∫(2x² - 5) dx from x = 3 to x = 1

Question 53

Which technique can be applied to integrate the function ∫x sin(x) dx?

Accepted Answer

Integration by parts

Answer

Integration by substitution

Answer

Integration by power rule

Answer

Integration by trigonometric identities

Question 54

State the general form of the antiderivative of the function f(x) = e^(2x).

Accepted Answer

e^(2x) + C

Answer

2e^(2x) + C

Answer

1/2e^(2x) + C

Answer

xe^(2x) + C

Question 55

Which of the following improper integrals converges?

Accepted Answer

∫[0,∞] e^(-x²) dx

Answer

∫[0,∞] 1/x² dx

Answer

∫[0,1] 1/√x dx

Answer

∫[-1,1] 1/(x-1) dx

Question 56

Which condition ensures the integrability of a function f(x) over the interval [a,b]?

Accepted Answer

f(x) is continuous at every point in [a,b]

Answer

f(x) has continuous first derivative in [a,b]

Answer

f(x) is bounded on [a,b]

Answer

f(x) is defined at every point in [a,b]

Question 57

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

Accepted Answer

1/2

Answer

1

Answer

-1/2

Answer

0

Question 58

Evaluate the definite integral ∫[1, 2] x^2 + 2x dx:

Accepted Answer

15/2

Answer

9/2

Answer

21/2

Answer

27/2

Question 59

Which technique is most suitable for integrating the function 1/(x^3 + a^3)?

Accepted Answer

Partial fractions

Answer

Integration by parts

Answer

Trigonometric substitution

Answer

Logarithmic integration

Question 60

Which of the following is not a fundamental rule of integration?

Accepted Answer

∫f'(x) dx = f(x)

Answer

∫[u + v] dx = ∫u dx + ∫v dx

Answer

∫c f(x) dx = c ∫f(x) dx

Answer

∫x^n dx = x^(n+1)/(n+1)

Question 61

Identify the integration technique suitable for evaluating ∫√(x^2 + 4) dx

Accepted Answer

Trigonometric substitution

Answer

Partial fractions

Answer

Logarithmic integration

Answer

Integration by parts