Master Integral Calculus: Online Quizzes for Pharmacy Students

Question 1

Find the indefinite integral of f(x) = x * sin(x) using integration by parts.

Accepted Answer

cos(x) - x * cos(x)

Answer

x * cos(x) + sin(x)

Answer

-x * sin(x) - cos(x)

Answer

sin(x) + C

Question 2

Which of the following techniques is used for integrating rational functions?

Accepted Answer

Partial fraction decomposition

Answer

Substitution

Answer

Integration by parts

Answer

Trigonometric substitution

Question 3

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

Accepted Answer

e^(2x)/2 + C

Answer

2e^(2x) + C

Answer

e^(2x) + C

Answer

x * e^(2x) + C

Question 4

Which of the following techniques is used to integrate trigonometric functions?

Accepted Answer

Trigonometric substitution

Answer

Integration by parts

Answer

Substitution

Answer

Partial fraction decomposition

Question 5

Evaluate the definite integral ∫[0, 1] (2x + 1) dx

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 6

Which of the following integration techniques is appropriate to evaluate ∫(x^3 - 2x^2 + 5x - 1) dx?

Accepted Answer

U-substitution

Answer

Integration by parts

Answer

Substitution

Answer

Trigonometric substitution

Question 7

Find the indefinite 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 8

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

Accepted Answer

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

Answer

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

Answer

ln(x^2 + 1) + C

Answer

ln|x^2 + 1| + C

Question 9

Which method is NOT a technique for finding integrals?

Accepted Answer

Logarithmic differentiation

Answer

Substitution

Answer

Integration by parts

Answer

Partial fractions

Question 10

The definite integral of f(x) over the interval [a, b] is denoted as:

Accepted Answer

∫[a,b] f(x) dx

Answer

∫[b,a] f(x) dx

Answer

∫[a-b] f(x) dx

Answer

∫[b-a] f(x) dx

Question 11

The integral of ∫(e^(2x) + sin(3x)) dx is:

Accepted Answer

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

Answer

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

Answer

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

Answer

2(e^(2x)) - 3(cos(3x)) + C

Question 12

Which property describes the behavior of the definite integral under addition of its subintervals?

Accepted Answer

Additivity

Answer

Linearity

Answer

Boundedness

Answer

Continuity

Question 13

To evaluate the integral ∫(x/(sqrt(1-x^2))) dx using substitution, the appropriate substitution is:

Accepted Answer

u = sqrt(1-x^2)

Answer

u = 1-x^2

Answer

u = x^2

Answer

u = x

Question 14

An example of an improper integral is:

Accepted Answer

∫[1,∞] e^(-x) dx

Answer

∫[0,1] (1/x^2) dx

Answer

∫[-π/2,π/2] sin(x) dx

Answer

∫[0,1] x dx

Question 15

Evaluate the indefinite integral: ∫x^2 e^x dx

Accepted Answer

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

Answer

x e^x - e^x + C

Answer

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

Answer

e^x / x^2 + C

Question 16

Find the area of the region bounded by the curve y = 2x^2 - 3 and the x-axis between x = -1 and x = 2.

Accepted Answer

14/3

Answer

4

Answer

16/3

Answer

2

Question 17

Find the integral: ∫(ln(x) + x^2) dx

Accepted Answer

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

Answer

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

Answer

ln(x)^2 + C

Answer

1 / x + x^2 + C

Question 18

Which of the following integrals represents the volume of the solid generated by rotating the region bounded by y = x and y = 2 about the y-axis?

Accepted Answer

∫[0,2] π(2^2 - x^2) dx

Answer

∫[0,2] 2π(2 - x)^2 dx

Answer

∫[0,2] 2πx(2 - x) dx

Answer

∫[0,2] πx^2 dx

Question 19

Find the indefinite integral of 3x^2 + 2x - 1.

Accepted Answer

x^3 + x^2 - x + C

Answer

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

Answer

6x + 2 + C

Answer

x^3 + x^2 - 1 + C

Question 20

Evaluate the definite integral of ∫(2x + 1) dx from 0 to 2.

Accepted Answer

6

Answer

2

Answer

4

Answer

8

Question 21

Which of the following techniques can be used to integrate the function ∫(x^2 + 1)/(x^3 + x) dx?

Accepted Answer

Substitution method

Answer

Partial fractions

Answer

Trigonometric substitution

Answer

Integration by parts

Question 22

Using integration by parts, find the integral of ∫x*sin(x) dx.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 23

The area bounded by the curve y = x^2, the x-axis, and the lines x = 1 and x = 2 can be calculated using which definite integral?

Accepted Answer

∫(from 1 to 2) x^2 dx

Answer

∫(from 0 to 1) x^2 dx

Answer

∫(from 0 to 2) x^2 dx

Answer

∫(from 1 to 2) x dx

Question 24

What is the integral of the function f(x) = 1/x?

Accepted Answer

ln|x| + C

Answer

x + C

Answer

-1/x^2 + C

Answer

1 + C

Question 25

Calculate the integral of ∫(1/(x^2 + 1)) dx.

Accepted Answer

arctan(x) + C

Answer

x/(x^2 + 1) + C

Answer

1/(x^2 + 1) + C

Answer

ln(x^2 + 1) + C

Question 26

The indefinite integral of a velocity function represents:

Accepted Answer

Displacement

Answer

Time

Answer

Acceleration

Answer

Speed

Question 27

Use partial fractions to evaluate the integral ∫(2x + 1)/(x^2 - 1) dx.

Accepted Answer

ln|x + 1| + ln|x - 1| + C

Answer

ln|x + 1| - ln|x - 1| + C

Answer

ln|2x + 1| + C

Answer

ln|x^2 - 1| + C