Master Integral Calculus: Conquer the Indefinite Integral

Question 1

Find the indefinite integral ∫sin(x) dx:

Accepted Answer

cos(x) + C

Answer

-cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Question 2

Which substitution should be used to evaluate the integral ∫√(x^2 + 1) dx?

Accepted Answer

u = √(x^2 + 1)

Answer

u = x^2 + 1

Answer

u = x

Answer

u = 1/x

Question 3

What is the indefinite integral of the function f(x) = sec^2(x)?

Accepted Answer

tan(x) + C

Answer

sec(x) + C

Answer

tan(x) - C

Answer

sec(x) - C

Question 4

Find the indefinite integral ∫(x^2 + 2x - 1)/(x + 1) dx:

Accepted Answer

x^2 - x + C

Answer

x^2 + x + C

Answer

x^2 - 2x + C

Answer

x^2 + 2x + C

Question 5

Which of the following integrals is an improper integral?

Accepted Answer

∫e^x dx from 0 to ∞

Answer

∫(1/x) dx from 0 to 1

Answer

∫sin(x) dx from -π/2 to π/2

Answer

∫x^2 dx from -1 to 1

Question 6

Find the indefinite integral of the vector-valued function F(t) = (t^2 + 1) i + (2t - 3) j.

Accepted Answer

(t^3/3 + t) i + (t^2 - 3t) j + C

Answer

t^2 i + 2t j + C

Answer

(3t^2 + 1) i + (2t - 3) j + C

Answer

(t^3 + 1) i + (t^2 - 3t) j

Question 7

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

Accepted Answer

x^3/3 + C

Answer

x^3 + C

Answer

3x^2 + C

Answer

2x^3 + C

Question 8

Determine the indefinite integral of sin(x):

Accepted Answer

cos(x) + C

Answer

-cos(x) + C

Answer

sin(x) + C

Answer

-sin(x) + C

Question 9

Use the substitution rule to calculate: ∫(x^2 + 1)^(1/2) dx

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 10

Calculate the area bounded by the curve y = x^2 between x = 0 and x = 2 using integration:

Accepted Answer

8/3

Answer

4/3

Answer

2/3

Answer

1/3

Question 11

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 12

Evaluate the integral ∫(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 13

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

Accepted Answer

ln(x^2 + 1) + C

Answer

√(x^2 + 1) + C

Answer

ln|x| + C

Answer

x^2 / √(x^2 + 1) + C

Question 14

Evaluate the integral ∫(1 / x^2 - 1) dx.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 15

Which of the following is the indefinite integral of f(x) = cos^2(x)?

Accepted Answer

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

Answer

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

Answer

sin(x) cos(x) + C

Answer

sin^2(x) + C

Question 16

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

Accepted Answer

cos(x) + C

Answer

-cos(x) + C

Answer

sin(x) + C

Answer

xcos(x) + C

Question 17

Determine the indefinite integral of e^x.

Accepted Answer

e^x + C

Answer

e^(x+1) + C

Answer

x e^x + C

Answer

x^2 e^x + C

Question 18

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

Accepted Answer

tan(x) + x + C

Answer

tan(x) - x + C

Answer

sec(x) + C

Answer

sec(x) - C

Question 19

Integrate (x + 1)/(x^2 - 1) with respect to x.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 20

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

Accepted Answer

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

Answer

1/(x^2+1) + C

Answer

arctan(x) + C

Question 21

Which of the following is the indefinite integral of ∫(sin^2(x)) dx?

Accepted Answer

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

Answer

2sin(x)cos(x) + C

Answer

sin^2(x) + C

Answer

-cos(x) + C

Question 22

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

Accepted Answer

1/2 x^3 e^(2x) - 3/4 x^2 e^(2x) + 3/8 x e^(2x) - 3/16 e^(2x) + C

Answer

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

Answer

x^4 e^(2x) + C

Question 23

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

Accepted Answer

tan(x) + C

Answer

sec(x) + C

Answer

1/sec(x) + C

Answer

-tan(x) + C

Question 24

Which of the following is the indefinite integral of ∫(cos(x) / sin^2(x)) dx?

Accepted Answer

-cot(x) + C

Answer

tan(x) + C

Answer

sec(x) + C

Question 25

Find the indefinite integral of the function ∫(e^(x) / (1 + e^(x))) dx.

Accepted Answer

ln(|1 + e^(x)|) + C

Answer

-ln(|1 + e^(x)|) + C

Answer

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

Answer

e^(x) - 1 + C

Question 26

Find the indefinite integral of f(x) = 2x + 3.

Accepted Answer

x^2 + 3x + C

Answer

x^2 - 3x

Answer

2x - 3x

Answer

2x^2 + 3x

Question 27

Evaluate the indefinite integral of f(x) = sin(x).

Accepted Answer

-cos(x) + C

Answer

cos(x)

Answer

sin(x)

Answer

tan(x)

Question 28

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

Accepted Answer

e^x + C

Answer

e^(x+1)

Answer

ln(x)

Answer

x*e^x

Question 29

Evaluate the indefinite integral of f(x) = 1/(x^2+1).

Accepted Answer

arctan(x) + C

Answer

ln(x^2 + 1)

Answer

1/(x+1)

Answer

tan(x)

Question 30

Find the indefinite integral of f(x) = x*sin(x).

Accepted Answer

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

Answer

x*cos(x) - sin(x)

Answer

x*cos(x) + sin(x)

Answer

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

Question 31

Evaluate the indefinite integral of f(x) = 1/x.

Accepted Answer

ln|x| + C

Answer

-ln|x|

Answer

x

Answer

x^-1

Question 32

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

Accepted Answer

tan(x) + C

Answer

sec(x)

Answer

cosec^2(x)

Answer

cot(x)

Question 33

Find the indefinite integral of f(x) = exp(2x+1).

Accepted Answer

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

Answer

(2/1)exp(2x+1)

Answer

-exp(2x+1)

Answer

exp(2x+1)

Question 34

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

Accepted Answer

e^x + C

Answer

x * e^x + C

Answer

e^(x+1) + C

Answer

e^x - 1 + C

Question 35

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

Accepted Answer

sin(x) + C

Answer

x * cos(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C

Question 36

To evaluate ∫sin(2x) dx, what is the most appropriate substitution?

Accepted Answer

u = 2x

Answer

u = sin(x)

Answer

u = cos(x)

Answer

u = tan(x)

Question 37

What is the indefinite integral of (x² + 5x - 3) with respect to x?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 38

What is the indefinite integral of (e^(2x) + sin(x)) with respect to x?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 39

Determine the indefinite integral of ∫(sin(2x)cos(2x)) dx.

Accepted Answer

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

Answer

cos^2(2x) + C

Answer

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

Answer

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

Question 40

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 41

Which of the following is the indefinite integral of ∫(1 / (x^2 - 1)) dx?

Accepted Answer

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

Answer

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

Answer

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

Answer

arctanh(x) + C

Question 42

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

Accepted Answer

(2 / 15)(x - 1)^(5 / 2) + C

Answer

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

Answer

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

Answer

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

Question 43

Find the indefinite integral of ∫(sin^n(x)cos(x)) dx, where n is a positive integer.

Accepted Answer

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

Answer

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

Answer

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

Answer

sin^n(x) + C