Master Integral Calculus: Online Quizzes and Practice Questions

Question 1

Calculate the area under the curve y = x^2 between the limits x = 0 and x = 2:

Accepted Answer

8/3

Answer

4/3

Answer

2/3

Answer

16/3

Question 2

Determine the centre of mass of a thin rod of uniform density ρ lying along the x-axis and of length L:

Accepted Answer

L/2

Answer

L/4

Answer

3L/4

Answer

L

Question 3

Which integration technique can be used to evaluate ∫x * e^x dx?

Accepted Answer

Integration by parts

Answer

Substitution

Answer

Partial fractions

Answer

Trigonometric substitution

Question 4

Identify the most appropriate substitution for integrating ∫(x^3 + 1)/(x^2 + 1) dx.

Accepted Answer

u = x^2 + 1

Answer

u = x^3 + 1

Answer

u = 1/(x^2 + 1)

Answer

u = x + 1

Question 5

Find the integral of (e^(2x) + 5x).

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 6

Identify the suitable substitution for integrating ∫√(x^2 + 1) dx.

Accepted Answer

u = √(x^2 + 1)

Answer

u = x^2 + 1

Answer

u = 1/√(x^2 + 1)

Answer

u = x + 1

Question 7

Which of the following is NOT a technique for integrating exponential functions?

Accepted Answer

Logarithmic differentiation

Answer

Substitution

Answer

Integration by parts

Answer

Partial fractions

Question 8

Find the indefinite integral of the function f(x) = 1 / (x^2 + 4).

Accepted Answer

1/2 arctan(x/2) + C

Answer

1/4 arctan(x/2) + C

Answer

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

Answer

-1/4 arctan(x/2) + C

Question 9

Which technique is most appropriate for integrating (x^2 + 2x + 3) / (x + 1) dx ?

Accepted Answer

Partial fractions

Answer

Long division

Answer

Integration by parts

Answer

Substitution

Question 10

Calculate the area under the curve y = x^2 - 2x + 1 between the points (0, 1) and (2, 1) :

Accepted Answer

2

Answer

0

Answer

4

Answer

6

Question 11

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

Accepted Answer

Independence of path

Answer

Additivity

Answer

Linearity

Answer

Continuity

Question 12

Which integral represents the work done by a force F(x) = 2x + 3 over the interval [0, 5] :

Accepted Answer

∫[0, 5] (2x + 3) dx

Answer

∫[5, 0] (2x + 3) dx

Answer

(2x + 3) ∫[0, 5] dx

Answer

(2x + 3) ∫[5, 0] dx

Question 13

Identify the appropriate substitution for integrating ∫√(x^2 + 1) dx :

Accepted Answer

u = √(x^2 + 1)

Answer

u = x

Answer

u = x^2

Answer

u = x^2 + 1

Question 14

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 15

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

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 16

Which of the following integration techniques would be most effective for ∫(x^2 + 1)/(x+1) dx ?

Accepted Answer

Partial fractions

Answer

Integration by parts

Answer

Trigonometric substitution

Answer

U-substitution

Question 17

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

Accepted Answer

sin(x)

Answer

-sin(x)

Answer

cos(x)

Answer

-cos(x)

Question 18

Which of the following is the correct integral for finding the center of mass of a thin rod with density ρ(x) = x over the interval [0, L]?

Accepted Answer

∫[0, L] xρ(x) dx

Answer

∫[0, L] ρ(x) dx

Answer

∫[0, L] x/ρ(x) dx

Answer

∫[0, L] ρ(x)/x dx

Question 19

Evaluate the integral ∫e^x(x + 1) dx.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 20

Which of the following is the correct substitution for integrating ∫√(x^2 + 1) dx?

Accepted Answer

u = √(x^2 + 1)

Answer

u = x

Answer

u = x^2 + 1

Answer

u = 1/√(x^2 + 1)

Question 21

Identify the appropriate substitution to integrate e^(2x+1).

Accepted Answer

u = 2x+1

Answer

u = e^(2x+1)

Answer

u = 2x

Answer

u = e^2x

Question 22

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

Accepted Answer

(2/3, 8/5)

Answer

(8/5, 2/3)

Answer

(3/2, 5/8)

Answer

(5/8, 3/2)

Question 23

Which condition must a function satisfy to be integrable over a closed interval?

Accepted Answer

Continuity

Answer

Differentiability

Answer

Boundedness

Answer

Definition at every point in the interval

Question 24

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

Accepted Answer

1

Answer

0

Answer

2

Answer

-1

Question 25

Find the integral of e^x.

Accepted Answer

e^x + C

Answer

e^x

Answer

(1/e)x + C

Answer

e^(x+C)

Question 26

Which of the following techniques is most appropriate for integrating the function f(x) = e^x * sin(x)? Select the correct option below.

Accepted Answer

Integration by parts

Answer

Substitution

Answer

Trigonometric substitution

Question 27

Calculate the center of mass of a uniform thin rod of length L and mass M. Select the correct answer from the options below.

Accepted Answer

L/2

Answer

L/4

Answer

L

Answer

3L/4

Question 28

Which of the following functions is NOT integrable? Choose the correct option from the list below.

Accepted Answer

f(x) = 1/x^2

Answer

f(x) = x^3

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 29

Which of the following is the correct formula for the trapezoidal rule? Choose the correct option from the list below.

Accepted Answer

∫[a, b] f(x) dx ≈ (b - a)/2 * [f(a) + f(b)]

Answer

∫[a, b] f(x) dx ≈ (b - a) * [f(a) - f(b)]

Answer

∫[a, b] f(x) dx ≈ (b - a) * [f(a) + f(b)]

Answer

∫[a, b] f(x) dx ≈ (b - a)/2 * [f(a) - f(b)]

Question 30

What is the indefinite integral of $3x^2 + 2x - 1$ with respect to $x$?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 31

Find $\int (2e^x + \frac{1}{x})\, dx$.

Accepted Answer

$2e^x + ln|x| + C$

Answer

$2e^{2x} + ln|x| + C$

Answer

$e^x + ln|x| + C$

Answer

$2e^x + \frac{1}{x^2} + C$

Question 32

Calculate the definite integral $\int_1^3 (4x - 2)\, dx$.

Accepted Answer

12

Answer

10

Answer

16

Answer

14

Question 33

What is the area bounded by the curve $y = x^2$, the x-axis, and the lines $x = 0$ and $x = 2$?

Accepted Answer

$\frac{8}{3}$

Answer

4

Answer

2

Answer

$\frac{4}{3}$

Question 34

The velocity of a particle moving along a straight line is given by $v(t) = 3t^2 - 6t + 2$ m/s. What is the displacement of the particle from $t = 1$ s to $t = 3$ s?

Accepted Answer

4 m

Answer

6 m

Answer

2 m

Answer

8 m

Question 35

Determine $\int \sin(2x) \, dx$.

Accepted Answer

$-\frac{1}{2} \cos(2x) + C$

Answer

$\frac{1}{2} \cos(2x) + C$

Answer

$\cos(2x) + C$

Answer

$-2 \cos(2x) + C$

Question 36

Find the volume of the solid generated when the region bounded by the curve $y = \sqrt{x}$, the x-axis, and the line $x = 4$ is rotated 360° about the x-axis.

Accepted Answer

8π

Answer

32π

Answer

16π

Answer

4π

Question 37

If $\frac{dy}{dx} = 6x^2 + 4x$ and the curve passes through the point (1, 5), find the equation of the curve.

Accepted Answer

$y = 2x^3 + 2x^2 + 1$

Answer

$y = 3x^3 + 2x^2 + 0$

Answer

$y = 6x^3 + 4x^2 - 7$

Answer

$y = 2x^3 + 2x^2 + 5$

Question 38

Evaluate $\int \frac{x}{x^2 + 1}\, dx$.

Accepted Answer

$\frac{1}{2} \ln(x^2 + 1) + C$

Answer

$\tan^{-1}(x) + C$

Answer

$ln(x^2 + 1) + C$

Answer

$\frac{1}{2} \tan^{-1}(x) + C$

Question 39

A region R is bounded by the curve $y = e^x$, the x-axis, the y-axis and the line $x = 2$.  Find the x-coordinate of the centroid of R.

Accepted Answer

$\frac{e^2 + 1}{e^2 - 1}$

Answer

$\frac{2e^2}{e^2 - 1}$

Answer

$\frac{e^2}{e^2 - 1}$

Answer

$\frac{1}{e^2 - 1}$

Question 40

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

Accepted Answer

-cos(x) + C

Answer

-sin(x) + C

Answer

sin(x) + C

Answer

cos(x) + C

Question 41

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

Accepted Answer

1

Answer

π/2

Answer

π

Answer

0

Question 42

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

Accepted Answer

∫udv = uv - ∫vdu

Answer

∫udv = du - ∫vdu

Answer

∫dudv = uv + ∫vdv

Answer

∫dudv = uv - ∫vdv

Question 43

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

Accepted Answer

(32π/5) cubic units

Answer

(8π/5) cubic units

Answer

(16π/5) cubic units

Answer

(4π/5) cubic units

Question 44

Which of the following is the most efficient method for integrating ∫(x^3 + 2x^2 + 1) dx?

Accepted Answer

Use the power rule.

Answer

Use substitution.

Answer

Use integration by parts.

Answer

Use partial fractions.

Question 45

Find the center of mass of a thin rod lying along the x-axis from x = 0 to x = L, with density function ρ(x) = k.

Accepted Answer

L/2

Answer

3L/4

Answer

L/4

Answer

L

Question 46

Which of the following integrals represents the area of the region bounded by the curves y = x^2 and y = 2x + 1?

Accepted Answer

∫[0,1] (2x + 1 - x^2) dx

Answer

∫[1,2] (2x + 1 - x^2) dx

Answer

∫[0,1] (x^2 - 2x - 1) dx

Question 47

Which of the following is the antiderivative of 1 / (x^2 + 1)?

Accepted Answer

arctan(x) + C

Answer

tan(x) + C

Answer

1 / (2x) + C

Answer

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

Question 48

Find the integral of ∫[0,∞] e^(-x^2) dx.

Accepted Answer

(π/2)^(1/2)

Answer

0

Answer

π/2

Answer

1 / (2π)^(1/2)