Master Integration in Calculus: Online Quizzes and Practice

Question 1

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

Accepted Answer

ln(|x - 1|) + C

Answer

-ln(|x - 1|) + C

Answer

1/(x - 1) + C

Answer

-1/(x - 1) + C

Question 2

Evaluate the integral: ∫(x^2 + 3x - 2) dx

Accepted Answer

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

Answer

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

Answer

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

Answer

(x^3)/3 + C

Question 3

Which trigonometric function has an integral involving the inverse trigonometric function arctan(x)?

Accepted Answer

tan(x)

Answer

sin(x)

Answer

cos(x)

Answer

cot(x)

Question 4

To calculate the area under the curve y = √(x) between x = 0 and x = 4, which definite integral should be evaluated?

Accepted Answer

∫[0,4] √(x) dx

Answer

∫[4,0] √(x) dx

Answer

∫[0,4] x² dx

Answer

∫[4,0] x² dx

Question 5

Which integration technique is most suitable for finding the integral of f(x) = e^(2x)?

Accepted Answer

u-substitution

Answer

Integration by parts

Answer

Partial fractions

Answer

None of these

Question 6

Which of the following functions is an antiderivative of f(x) = 1/x?

Accepted Answer

ln(x)

Answer

1/x²

Answer

e^x

Answer

x²

Question 7

Find the integral of ∫(cos(x) - sin(x))² dx.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 8

Which statement is NOT included in the fundamental theorem of calculus?

Accepted Answer

If f(x) is differentiable on an open interval, then f'(x) = d/dx ∫f(x) dx

Answer

If f(x) is continuous on [a, b], then ∫[a,b] f(x) dx = F(b) - F(a), where F'(x) = f(x)

Answer

Every differentiable function on a closed interval has an antiderivative

Answer

The derivative of an integral with respect to its upper limit is the integrand

Question 9

Which technique is NOT used for integration?

Accepted Answer

Quadratic formula

Answer

U-substitution

Answer

Integration by parts

Answer

Trigonometric identities

Question 10

Evaluate the integral ∫cos^2(3x) dx.

Accepted Answer

x/2 + sin(6x)/12 + C

Answer

x/2 - sin(6x)/12 + C

Answer

x/2 + sin(6x)/6 + C

Answer

x/2 - sin(6x)/6 + C

Question 11

Find the integral of the polynomial (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^3/2 + x^2 - 2x + C

Answer

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

Question 12

Determine the volume of the solid generated by rotating the region bounded by y = x^2 and y = 4 about the x-axis:

Accepted Answer

16π

Answer

8π

Answer

32π

Answer

64π

Question 13

Find the integral of the product (x * ln(x)) with respect to x:

Accepted Answer

(x^2/2) * ln(x) - (x^2/4) + C

Answer

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

Answer

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

Answer

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

Question 14

Select the appropriate substitution to evaluate the integral ∫[0,1] x * sqrt(1 - x^2) dx:

Accepted Answer

u = 1 - x^2

Answer

u = sqrt(1 - x^2)

Answer

u = x

Answer

u = x^2

Question 15

Which of the following is an incorrect method for integration?

Accepted Answer

Logarithmic differentiation

Answer

Integration by substitution

Answer

Integration by parts

Answer

Trigonometric substitution

Question 16

Find the integral of x² + 2x - 1 with respect to x.

Accepted Answer

x³/3 + x² - x + C

Answer

x³ + 2x² - x + C

Answer

(x² + 2x - 1)/2 + C

Answer

2x³ + 4x² - 2x + C

Question 17

Find the integral of ∫√(x + 1) dx.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 18

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

Accepted Answer

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

Answer

x^3 + x^2 - x + C

Answer

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

Answer

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

Question 19

Which of the following techniques can be used to integrate ∫e^(2x) cos x dx?

Accepted Answer

Integration by parts

Answer

Substitution

Answer

Trigonometric identities

Answer

Integration by partial fractions

Question 20

What is the order of integration for the double integral ∬f(x, y) dx dy?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 21

Find the average value of the function f(x) = x² on the interval [0, 2].

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 22

Calculate the antiderivative of (2x + 5)e^(x^2 + 5x) with respect to x.

Accepted Answer

2e^(x^2 + 5x) + 5xe^(x^2 + 5x) + C

Answer

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

Answer

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

Answer

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

Question 23

Which of the following integrals represents the volume of the solid generated by rotating the region bounded by the curves y = x^3 and y = 8 about the x-axis?

Accepted Answer

∫[0, 2] (8^3 - x^9) dx

Answer

∫[0, 2] (x^3 - 8^3) dx

Answer

∫[0, 2] (8x - x^3) dx

Answer

∫[0, 2] (x^9 - 8^3) dx

Question 24

Which of the following is a general form of the antiderivative of cos(x) / sin^3(x)?

Accepted Answer

-1/2 cot(x) + C

Answer

1/2 cot(x) + C

Answer

-1/sin^2(x) + C

Answer

1/sin^2(x) + C

Question 25

Determine the volume of the solid formed by rotating the region bounded by y = x^2 and y = 4 about the x-axis:

Accepted Answer

128π/15

Answer

64π/5

Answer

32π/5

Answer

16π/5

Question 26

Calculate the integral of e^x:

Accepted Answer

e^x + C

Answer

e^x/x + C

Answer

xe^x + C

Answer

x^e + C

Question 27

Find the indefinite integral of (x^2 + e^x).

Accepted Answer

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

Answer

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

Answer

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

Answer

(x^3 / 3) + e^x

Question 28

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

Accepted Answer

1

Answer

0

Answer

-1

Answer

π/2

Question 29

Select the integration technique that can be used to solve the integral ∫(x^2 + 1)/(x + 1) dx.

Accepted Answer

Partial fractions

Answer

Long division

Answer

Trigonometric substitution

Answer

Integration by parts

Question 30

Which of the following techniques is appropriate for integrating ∫(x^2 + 1)/(x+1) dx:

Accepted Answer

Partial Fractions

Answer

Substitution

Answer

Integration by Parts

Answer

Trig Substitution

Question 31

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

Accepted Answer

sin(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C

Answer

-cos(x) + C

Question 32

Calculate the volume of the solid generated by rotating the region bounded by y = x^2 and y = 4 about the x-axis.

Accepted Answer

π∫(4 - x^2)^2dx

Answer

π∫(x^2 - 4)^2dx

Answer

2π∫(4 - x^2)^2dx

Answer

2π∫(x^2 - 4)^2dx

Question 33

State the integration-by-parts formula:

Accepted Answer

∫udv = uv - ∫vdu

Answer

∫udu = u^2/2 + C

Answer

∫e^udu = e^u + C

Answer

∫sin(u)du = -cos(u) + C

Question 34

Calculate the indefinite integral of the function f(x) = x^2 + 3x - 5.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 35

Find the integral of the function f(x) = sin(2x) with respect to x.

Accepted Answer

-cos(2x) + C

Answer

cos(2x) + C

Answer

-cos(x) + C

Answer

cos(x) + C

Question 36

Which of the following is not considered a property of definite integrals?

Accepted Answer

Chain rule

Answer

Linearity

Answer

Additivity

Answer

Power rule

Question 37

An object moves along a straight line with a velocity of v(t) = t^2 - 4t + 5 meters per second. What is the displacement of the object over the time interval [0, 3]?

Accepted Answer

9.5 meters

Answer

5.5 meters

Answer

-9.5 meters

Question 38

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

Accepted Answer

Differentiation

Answer

Substitution

Answer

Trigonometric identities

Answer

Integration by parts

Question 39

Evaluate the integral ∫(x^2 + e^x) dx.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 40

Which of the following is the correct formula for the integral of the trigonometric function sin(x)?

Accepted Answer

-cos(x) + C

Answer

sin(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C