Integrals: Master the Art of Finding Areas Under Curves

Question 1

Which of the following is the integral of the function f(x) = 2x + 1?

Accepted Answer

2x + C

Answer

x^2 + x + C

Answer

2x^2 + 1

Answer

x^2 + C

Question 2

Which of the following is the integral of the function f(x) = e^x?

Accepted Answer

e^x + C

Answer

e^x - C

Answer

x * e^x + C

Answer

1/e^x + C

Question 3

An object's velocity is given by the function v(t) = 2t + 1. Calculate its displacement over the time interval from t = 0 to t = 2.

Accepted Answer

8

Answer

6

Answer

10

Answer

12

Question 4

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

Accepted Answer

1 / 2 * cos(2x) + C

Answer

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

Answer

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

Answer

1 / 2 * sin(2x) + C

Question 5

Find the integral of f(x) = x^2.

Accepted Answer

x^3/3 + C

Answer

x^4/4 + C

Answer

x^5/5 + C

Answer

x^6/6 + C

Question 6

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

Accepted Answer

1

Answer

0

Answer

2

Answer

-1

Question 7

Integrate the polynomial: ∫(x^3 + 2x^2 - 3x + 4) dx.

Accepted Answer

(x^4)/4 + (2x^3)/3 - 3(x^2)/2 + 4x + C

Answer

(x^4)/4 - (2x^3)/3 + 3(x^2)/2 + 4x + C

Answer

(x^4)/4 - (2x^3)/3 - 3(x^2)/2 + 4x + C

Answer

(x^4)/4 + (2x^3)/3 - 3(x^2)/2 - 4x + C

Question 8

Define the integral of a function.

Accepted Answer

The limit of a sum of infinitesimally small areas

Answer

The area enclosed by the graph of the function and the x-axis

Answer

The derivative of the function

Answer

The rate of change of the function

Question 9

To determine the area under the curve of the function f(x) = x^2 between x = 0 and x = 2, which integral should be used?

Accepted Answer

∫[0, 2] x^2 dx

Answer

∫[2, 0] x^2 dx

Answer

∫[0, ∞] x^2 dx

Answer

∫[-∞, 0] x^2 dx

Question 10

Which of the following properties does NOT apply to integrals?

Accepted Answer

The integral of a product of functions is the product of the integrals of the functions

Answer

Integrals are linear

Answer

The integral of a constant multiplied by a function is the constant multiplied by the integral of the function

Answer

The integral of a sum of functions is the sum of the integrals of the functions

Question 11

Which integration technique is appropriate for finding the integral of (x^2 + 1)/(x + 1)?

Accepted Answer

Partial fractions

Answer

Integration by parts

Answer

Trigonometric substitution

Answer

U-substitution

Question 12

The integral ∫[0, π/2] cos(x) dx represents which of the following?

Accepted Answer

The area under the cosine curve from 0 to π/2

Answer

The volume of the solid generated by rotating the cosine curve from 0 to π/2 around the x-axis

Answer

The length of the cosine curve from 0 to π/2

Answer

The average value of the cosine function from 0 to π/2

Question 13

To evaluate the integral ∫x^2 * ln(x) dx, which of the following techniques can be used?

Accepted Answer

Integration by parts

Answer

U-substitution

Answer

Trigonometric substitution

Answer

Partial fractions

Question 14

What is the order of the differential equation y'' + y = 0?

Accepted Answer

Second order

Answer

First order

Answer

Third order

Answer

Fourth order

Question 15

Find the solution to the differential equation y' - 2y = 0.

Accepted Answer

y = e^-2x

Answer

y = e^2x

Answer

y = 2e^x

Answer

y = 2e^-x

Question 16

Which of the following is a common application of the Laplace transform?

Accepted Answer

Solving differential equations

Answer

Finding the inverse Laplace transform

Answer

Evaluating integrals

Answer

Solving partial differential equations

Question 17

The definite integral of sin(x) from 0 to π/2 is:

Accepted Answer

(b) 1

Answer

(a) 0

Answer

(c) -1

Answer

(d) 2

Question 18

The integral of e^(2x+1)dx is:

Accepted Answer

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

Answer

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

Answer

(c) (1/8)e^(2x+1) + C

Answer

(d) (1/16)e^(2x+1) + C

Question 19

When evaluating definite integrals using the method of substitution, which of the following steps is NOT required?

Accepted Answer

Apply integration by parts to evaluate the resulting expression.

Answer

Find the derivative of the integrand.

Answer

Substitute the expression for the new variable into the integral.

Answer

Change the limits of integration accordingly.

Question 20

Find the indefinite integral of x^2 + 2x + 1.

Accepted Answer

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

Answer

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

Answer

x^2 + 2x + C

Answer

x^3/2 + 2x + C

Question 21

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

Accepted Answer

1

Answer

2

Answer

-1

Answer

0

Question 22

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

Accepted Answer

arctan(x) + C

Answer

x/(x^2 + 1) + C

Answer

1/x + C

Answer

ln(x^2 + 1) + C

Question 23

Determine the area under the curve of y = x^2 + 1 between x = 0 and x = 2.

Accepted Answer

9

Answer

5

Answer

16

Answer

4

Question 24

Select the appropriate integration technique to evaluate the integral of sin(x)cos(x).

Accepted Answer

Integration by substitution

Answer

Integration by trigonometric substitution

Answer

Integration by partial fractions

Answer

Integration by parts

Question 25

Find the integral of x^n with respect to x, where n is a positive integer.

Accepted Answer

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

Answer

x^(n-1) + C

Answer

(n+1)x^n + C

Answer

x^n + C

Question 26

Find the integral of e^x.

Accepted Answer

e^x + C

Answer

ln(e^x) + C

Answer

e^(x+1) + C

Answer

x^e + C

Question 27

Find the indefinite integral of cot(x).

Accepted Answer

ln|sin(x)| + C

Answer

sin(x) + C

Answer

tan(x) + C

Answer

-csc(x) + C

Question 28

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

Accepted Answer

2√x + C

Answer

x/√x + C

Answer

1/x + C

Answer

√x + C

Question 29

Find the indefinite integral of sec^2(x).

Accepted Answer

tan(x) + C

Answer

csc(x) + C

Answer

cot(x) + C

Answer

sec(x) + C

Question 30

Which of the following is the antiderivative of f(x) = x^2?

Accepted Answer

x^3/3 + C

Answer

x^4/4 + C

Answer

2x + C

Answer

x + C

Question 31

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

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Answer

xcos(x) + C

Question 32

Which of the following is the Fundamental Theorem of Calculus?

Accepted Answer

The derivative of the integral of a function is the function itself.

Answer

The integral of the derivative of a function is the function itself.

Answer

The derivative of an integral is the integrand.

Question 33

What is the area under the curve of f(x) = x^2 + 1 from x = 0 to x = 2?

Accepted Answer

9/3

Answer

7/3

Answer

5/3

Answer

13/3

Question 34

What is the volume generated by rotating the region bounded by y = x and y = x^2 about the x-axis?

Accepted Answer

π/6

Answer

π/2

Answer

π/12

Question 35

What is the integral of ∫e^(2x) dx?

Accepted Answer

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

Answer

e^(2x) + C

Answer

2e^(2x) + C

Answer

e^(x) + C

Question 36

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

Accepted Answer

∫f(u)du = ∫f(x)dx * du/dx

Answer

∫f(u)du = ∫f(x)dx / du/dx

Answer

∫f(u)du = ∫f(x)dx / dx/du

Answer

∫f(u)du = ∫f(x)dx * dx/du

Question 37

What is the improper integral of ∫(1/x) dx from x = 1 to x = ∞?

Accepted Answer

∞

Answer

-∞

Answer

1

Answer

0

Question 38

Which of the following integrals represents the work done by a force F(x) acting on an object over a distance x?

Accepted Answer

∫F(x) dx

Answer

∫F'(x') dx

Answer

∫F(x') dx

Answer

∫F'(x) dx