Indefinite Integrals: Master the Basics with Interactive Quizzes

Question 1

What is the indefinite integral of sin(x)?

Accepted Answer

cos(x) + C

Answer

cos(x)

Answer

-cos(x) + C

Answer

sin(x) + C

Question 2

Which substitution would be most useful for evaluating the integral of (x^2 + 1)/(x+1)?

Accepted Answer

u = x + 1

Answer

u = x^2 + 1

Answer

u = x^2

Answer

u = 1/x

Question 3

What is the indefinite integral of e^(2x)?

Accepted Answer

e^(2x)/2 + C

Answer

2e^(2x) + C

Answer

e^(2x) + 1/2

Answer

x * e^(2x) + C

Question 4

Find the indefinite integral of ∫x * sqrt(x^2 + 1) dx.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 5

What is the indefinite integral of x^2?

Accepted Answer

x^3/3 + C

Answer

x^3 - C

Answer

2x + C

Answer

Cannot be determined

Question 6

Evaluate the indefinite integral of (e^x) / (1 + e^x) dx.

Accepted Answer

ln |1 + e^x| + C

Answer

-ln |1 + e^x| + C

Question 7

Find the indefinite integral of (1 / x) * ln x dx.

Accepted Answer

ln |ln x| + C

Answer

ln x / x + C

Question 8

Evaluate the indefinite integral of (x^3 - 2x^2 + 5x - 1) dx.

Accepted Answer

(1 / 4)x^4 - (2 / 3)x^3 + (5 / 2)x^2 - x + C

Answer

-x^4 + 2x^3 - 5x^2 + x + C

Question 9

Determine the indefinite integral of (tan x * sec^2 x) dx.

Accepted Answer

sec^2 x + C

Answer

tan^2 x + C

Question 10

Calculate the indefinite integral of the function q(x) = x * e^(-x²) with respect to x.

Accepted Answer

1/2 e^(-x²) + C

Answer

-1/2 e^(-x²) + C

Answer

-1/2 e^(x²) + C

Answer

1/2 e^(x²) + C

Question 11

Find the indefinite integral of f(x) = cot(x).

Accepted Answer

ln(|sin(x)|) + C

Answer

csc(x) + C

Answer

sec(x) + C

Answer

tan(x) + C

Question 12

Evaluate the indefinite integral of the square root of (1 + x^2) with respect to x.

Accepted Answer

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

Answer

(1/4) * (x * sqrt(1 + x^2) + ln(|x + sqrt(1 + x^2)|)) + C

Answer

(1/2) * (x * sqrt(1 + x^2) - ln(|x + sqrt(1 + x^2)|)) + C

Question 13

Find the indefinite integral of e^(2x + 1) with respect to x.

Accepted Answer

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

Answer

e^(2x + 2) + C

Answer

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

Answer

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

Question 14

Find the indefinite integral of cos^2(x) with respect to x.

Accepted Answer

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

Answer

1/4 * x + sin(x)cos(x) + C

Answer

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

Answer

1/4 * x - sin(x)cos(x) + C

Question 15

Which of the following is the correct indefinite integral of ∫(1/x) dx where x is greater than 0?

Accepted Answer

ln|x| + C

Answer

1/x² + C

Answer

x + C

Answer

-1/x² + C

Question 16

If F(x) is the antiderivative of f(x), what is the relationship between F'(x) and f(x)?

Accepted Answer

F'(x) = f(x)

Answer

F'(x) = -f(x)

Answer

F'(x) = f(x) + C

Answer

F'(x) = ∫f(x) dx

Question 17

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

Accepted Answer

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

Answer

2e^(2x) + C

Answer

e^(2x) + C

Question 18

Evaluate the indefinite integral of ∫(4x³ - 5x + 2) dx.

Accepted Answer

x⁴ - (5/2)x² + 2x + C

Answer

12x² - 5 + C

Answer

x⁴ - 5x² + 2x + C

Answer

4x⁴ - 5x² + 2x + C

Question 19

Find the indefinite integral of ∫(1/√(1-x²)) dx.

Accepted Answer

arcsin(x) + C

Answer

-arcsin(x) + C

Answer

arctan(x) + C

Answer

arccos(x) + C

Question 20

Find the indefinite integral of  ∫(x^2 + 2x - 1) dx.

Accepted Answer

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

Answer

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

Answer

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

Question 21

What is the indefinite integral of e^x dx?

Accepted Answer

e^x + C

Answer

x^2 e^x + C

Answer

x e^x + C

Question 22

Which of the following is the correct indefinite integral of ln(x) dx?

Accepted Answer

x ln(x) - x + C

Answer

ln(x^2) + C

Answer

x/ln(x) + C

Answer

ln(x) + C

Question 23

Find the indefinite integral of 1/(x+1) dx.

Accepted Answer

ln(|x+1|) + C

Answer

-ln(|x+1|) + C

Answer

(x+1) + C

Answer

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

Question 24

Find the indefinite integral of dx / (x^2 - 1).

Accepted Answer

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

Answer

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

Answer

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

Answer

ln(|x-1|) + C

Question 25

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

Accepted Answer

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

Answer

xsqrt(x^2 + 1) + C

Answer

sqrt(x^2 + 1) + C

Answer

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

Question 26

Which of the following is the indefinite integral of arctan(x) dx?

Accepted Answer

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

Answer

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

Answer

arctan(x) + C

Answer

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

Question 27

Evaluate $\int_{0}^{\pi/2} \sin(x) \cos(x) dx $.

Accepted Answer

$\frac{1}{2}$

Answer

$-1$

Answer

1

Answer

0

Question 28

If $\int f(x) dx = x^2 + \cos(x) + C$, what is $f(x)$?

Accepted Answer

$2x - \sin(x)$

Answer

$2x + \cos(x)$

Answer

$x^2 - \sin(x)$

Answer

$x^2 + \sin(x)$