Master Integral Calculus: Practice Quizzes and Exam Prep

Question 1

Find the integral of the function f(x) = x³ + 2x² - 1.

Accepted Answer

(x^4)/4 + (2x³)/3 - x + C

Answer

(x³)/3 + (2x²)/2 - x + C

Answer

(x^4)/4 + (2x³)/3 + x + C

Question 2

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

Accepted Answer

The area under a curve can be found using integration.

Answer

The derivative of an integral is the original function.

Answer

The integral of a derivative is the original function.

Answer

Integration can be used to find the volume of a solid of revolution.

Question 3

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

Accepted Answer

∫u dv = u v - ∫v du

Answer

∫u dv = v du - ∫u ddv

Answer

∫dv = u v + ∫v du

Answer

∫du = v du - ∫u ddv

Question 4

Which of the following is the correct formula for the arc length of a curve y = f(x) from x = a to x = b?

Accepted Answer

∫√(1 + (dy/dx)²) dx

Answer

∫(dy/dx) dx

Answer

∫√(dx² + dy²)

Answer

∫(dx² + dy²) dx

Question 5

Which of the following is the indefinite integral of (x² + 1) with respect to x?

Accepted Answer

½ x³ + x + C

Answer

x³ + C

Answer

x³ + x² + C

Answer

x² + 1 + C

Question 6

Which of the following is the integral of (1/x) dx?

Accepted Answer

ln(|x|) + C

Answer

x + C

Answer

ln(x) + C

Answer

x² + C

Question 7

Calculate the integral of the function h(x) = e^x.

Accepted Answer

e^x + C

Answer

xe^x + C

Answer

e^(x+1) + C

Answer

x + C

Question 8

Calculate the integral of the function g(x) = 1/(x^2 + 1).

Accepted Answer

arctan(x) + C

Answer

tan(x) + C

Answer

arctan(x^2) + C

Answer

tan(x^2) + C

Question 9

Which of the following integrals represents the area under the curve y = x² + 1 from x = 0 to x = 2?

Accepted Answer

∫[0, 2] (x² + 1) dx

Answer

∫[0, 2] (x³ + x)/2 dx

Answer

∫[0, 2] (2x² + 1) dx

Question 10

Which of the following is the correct substitution for evaluating the integral ∫(x² + 1)³ dx?

Accepted Answer

u = x² + 1

Answer

u = (x² + 1)⁴

Answer

u = x³ + 1

Question 11

Which of the following is the indefinite integral of tan(x)?

Accepted Answer

ln|cos(x)|

Answer

sec²(x)

Answer

-sec²(x)

Question 12

Evaluate the improper integral ∫[0, ∞] (1/(x² + 1)) dx.

Accepted Answer

π

Answer

1

Answer

∞

Question 13

Which of the following is a necessary condition for the integral ∫(f(x)/g(x)) dx to exist?

Accepted Answer

g(x) ≠ 0 for all x in the interval of integration

Answer

f(x) and g(x) are continuous

Answer

f(x) is nonnegative

Question 14

Find the indefinite integral of ∫e^(2x - 3) dx.

Accepted Answer

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

Answer

(1/2)e^(2x - 3) - C

Answer

e^(2x - 3) + C

Question 15

Which of the following is the most appropriate method for integrating ∫(x/(x² + 1)²) dx?

Accepted Answer

Integration by substitution

Answer

Integration by parts

Answer

Trigonometric substitution

Question 16

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

Accepted Answer

1

Answer

0

Answer

-1

Question 17

Which of the following is NOT a method of integration?

Accepted Answer

Chain rule

Answer

Power rule

Answer

Substitution rule

Answer

Integration by parts

Question 18

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 19

Which of the following integrals represents the area under the curve y = x² from x = 0 to x = 2?

Accepted Answer

∫[0,2] x² dx

Answer

∫[2,0] x² dx

Answer

∫[0,2] 2x dx

Answer

∫[2,0] 2x dx

Question 20

Which of the following is NOT a property of integrals?

Accepted Answer

∫∫f(x, y) dx dy = ∫∫f(y, x) dy dx

Answer

∫(u + v) dx = ∫u dx + ∫v dx

Answer

∫(cu) dx = c∫u dx

Answer

∫(1/u) dx = ln|u| + C

Question 21

The improper integral ∫(1/x) dx from 1 to ∞ is:

Accepted Answer

Converges to ln(2)

Answer

Diverges to infinity

Answer

Diverges to negative infinity

Answer

Converges to infinity

Question 22

Find the integral of the expression 3x² with respect to x.

Accepted Answer

x³ + C

Answer

9x³ + C

Answer

x² + C

Answer

6x + C

Question 23

Evaluate the integral of the function cos(2x) with respect to x.

Accepted Answer

½sin(2x) + C

Answer

sin(2x) + C

Answer

-½sin(2x) + C

Question 24

Calculate the definite integral of x² from the lower limit of 0 to the upper limit of 1.

Accepted Answer

1/3

Answer

1/2

Answer

1

Answer

2/3

Question 25

Determine the expression for the area under the curve of the function y = x³ between the points x = 0 and x = 2.

Accepted Answer

∫(from 0 to 2) x³ dx

Answer

∫(from 0 to 2) 3x² dx

Answer

∫(from 0 to 2) x dx

Answer

∫(from 0 to 2) x² dx

Question 26

Determine the volume of the solid generated by revolving the region bounded by the curve y = √x, the x-axis, and the vertical line x = 2 about the x-axis.

Accepted Answer

π∫(from 0 to 4) (√x)² dx

Answer

π∫(from 0 to 4) x dx

Answer

π∫(from 0 to 2) (√x)² dx

Answer

π∫(from 0 to 2) x dx

Question 27

Find the derivative of the function F(x) = ∫(from 0 to x) t² dt.

Accepted Answer

x²

Answer

t²

Answer

x³ / 3

Answer

2x

Question 28

Which of the following techniques is not typically employed in integration?

Accepted Answer

Factoring by grouping

Answer

Trigonometric substitution

Answer

Partial fractions

Answer

Integration by parts

Question 29

The integral of a constant function f(x) = C, where C is a constant, results in a function of what type?

Accepted Answer

A linear function

Answer

A quadratic function

Answer

Another constant function

Question 30

Find the integral of the function 1/x with respect to x.

Accepted Answer

ln|x| + C

Answer

x + C

Answer

x² / 2 + C

Answer

-1/x² + C

Question 31

Evaluate the improper integral ∫(from 1 to ∞) 1/x² dx.

Accepted Answer

1

Answer

-1

Answer

∞

Answer

0

Question 32

What is the integral of 3x² + 2x - 1 with respect to x?

Accepted Answer

x³ + x² - x + C

Answer

6x + 2 + C

Answer

9x⁴ + 4x² - 1 + C

Answer

x³ + x² - 1 + C

Question 33

Find the integral of ∫(2x + 1)³ dx.

Accepted Answer

¼(2x + 1)⁴ + C

Answer

3(2x + 1)² + C

Answer

½(2x + 1)⁴ + C

Answer

(2x + 1)⁴ + C

Question 34

The area under the curve y = x² from x = 0 to x = 2 is represented by which definite integral?

Accepted Answer

∫₀² x² dx

Answer

∫₀² √x dx

Answer

∫₀² x dx

Answer

∫₀² 2x dx

Question 35

Evaluate the definite integral ∫₁³ (x + 1) dx.

Accepted Answer

6

Answer

10

Answer

4

Answer

8

Question 36

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

Accepted Answer

-cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Question 37

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

Accepted Answer

e^x + C

Answer

xe^x + C

Answer

1/x + C

Answer

ln(x) + C

Question 38

Find the integral of ∫(1/x) dx.

Accepted Answer

ln|x| + C

Answer

x + C

Answer

-1/x² + C

Answer

1/x² + C

Question 39

The volume of the solid generated by revolving the region bounded by the curve y = x² and the x-axis from x = 0 to x = 2 about the x-axis is given by which integral?

Accepted Answer

π∫₀² (x²)² dx

Answer

π∫₀² x⁴ dx

Answer

2π∫₀² x⁴ dx

Answer

2π∫₀² x² dx

Question 40

If the velocity of a particle is given by v(t) = 2t + 1, what is the displacement of the particle from t = 0 to t = 3?

Accepted Answer

12

Answer

9

Answer

15

Answer

6

Question 41

The average value of the function f(x) = 3x² over the interval [0, 2] is:

Accepted Answer

4

Answer

6

Answer

8

Answer

2