Indefinite Integral: Definition, Formulas, and Techniques

Question 1

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

Accepted Answer

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

Answer

2x + ln(|x|) + C

Answer

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

Answer

ln(|x|) + C

Question 2

Which of the following is NOT a property of the indefinite integral?

Accepted Answer

∫f(x) dx = x^f(x) + C

Answer

∫cf(x) dx = c∫f(x) dx

Answer

∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

Answer

∫f(g(x)) dx = f(x)∫g'(x) dx

Question 3

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

Accepted Answer

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

Answer

(1/2)e^(2x) cos(x) + (1/2)e^(2x) sin(x)

Answer

e^(2x) cos(x)

Answer

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

Question 4

Find 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

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

Answer

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

Answer

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

Question 5

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

Accepted Answer

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

Answer

(1/3)e^(3x) + (1/3)x^3

Answer

e^(3x) + (1/3)x^3

Answer

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

Question 6

Which of the following is the precise definition of the indefinite integral of a function f(x)?

Accepted Answer

The family of all functions whose derivative is f(x).

Answer

The function whose derivative is f(x).

Answer

The area under the curve of f(x) between two points.

Answer

The total change of f(x) over a certain interval.

Question 7

Calculate the indefinite integral of ∫(x^2 + 3x - 1) dx:

Accepted Answer

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

Answer

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

Answer

2x^3 + 9x^2 - 3x + C

Answer

x^2 + 3x - 1 + C

Question 8

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

Accepted Answer

e^(2x)/2 + C

Answer

2e^(2x) + C

Answer

1/2e^(2x) + C

Answer

e^(2x) + C

Question 9

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

Accepted Answer

∫f(g(x))g'(x) dx = ∫f(u) du

Answer

∫f(g(x))g'(x) dx = ∫f(x) dx

Answer

∫f(g(x))g'(x) dx = u

Answer

∫f(g(x))g'(x) dx = f(g(x))

Question 10

Find the indefinite integral ∫sin(x)cos(x) dx:

Accepted Answer

1/2sin(2x) + C

Answer

sin(x)/2 + C

Answer

-cos(x) + C

Answer

cos(x)/2 + C

Question 11

Which of the following correctly represents the integral ∫(1/x) dx?

Accepted Answer

ln(|x|) + C

Answer

1/x + C

Answer

-1/x^2 + C

Answer

x + C

Question 12

Which statement about the indefinite integral is correct?

Accepted Answer

It can have multiple antiderivatives.

Answer

It always results in a unique function.

Answer

It is the inverse operation of differentiation.

Answer

It is always a polynomial function.

Question 13

Find the indefinite integral ∫√(x) dx:

Accepted Answer

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

Answer

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

Answer

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

Answer

x^(1/2) + C

Question 14

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

Accepted Answer

Integration by partial fractions

Answer

Integration by parts

Answer

Substitution

Answer

Integration by trigonometric substitution

Question 15

Find the indefinite integral of ∫(1 + sqrt(x)) dx with respect to x.

Accepted Answer

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

Answer

x + sqrt(x) + C

Answer

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

Answer

x * sqrt(x) + C

Question 16

Which of the following is NOT a step in the integration by substitution method?

Accepted Answer

Substitute the original expression back into the final answer.

Answer

Solve for x in terms of u.

Answer

Substitute u and du into the integral.

Answer

Integrate the resulting integral.

Question 17

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

Accepted Answer

1/2 * sin(2x) + C

Answer

sin(2x) + C

Answer

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

Answer

cos(2x) + C

Question 18

Which of the following is the indefinite integral of x^3 dx?

Accepted Answer

x^4/4 + C

Answer

x^2/2 + C

Answer

x^4 + C

Answer

x^3 + C

Question 19

Evaluate the indefinite integral of (2x + 5) dx:

Accepted Answer

x^2 + 5x + C

Answer

x^2 + 5 + C

Answer

2x^2 + 5x + C

Answer

2x^2 + 5 + C

Question 20

Which substitution should be used to integrate ∫(x^2 + 1)^(1/2) dx?

Accepted Answer

u = x^2 + 1

Answer

u = x

Answer

u = x^2

Answer

u = 1/2

Question 21

Find the indefinite integral of sin(x) cos(x) dx:

Accepted Answer

1/2 sin^2(x) + C

Answer

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

Answer

1/2 cos^2(x) + C

Answer

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

Question 22

Which of the following integrals represents the area under the curve y = x^2 from x = 0 to x = 2?

Accepted Answer

∫[0,2] x^2 dx

Answer

∫[2,0] x^2 dx

Answer

∫[0,2] 2x dx

Answer

∫[2,0] 2x dx

Question 23

Which of the following is NOT a technique for evaluating indefinite integrals?

Accepted Answer

Numerical integration

Answer

Power rule

Answer

u-substitution

Answer

Integration by parts

Question 24

Find the indefinite integral of sin^2(x)cos^2(x)dx:

Accepted Answer

(1/8)sin^4(x) + (1/8)cos^4(x) + C

Answer

(1/8)sin^4(x) - (1/8)cos^4(x) + C

Answer

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

Answer

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

Question 25

Find the indefinite integral of the function 3x^2.

Accepted Answer

x^3 + C

Answer

3x^3

Answer

3x + C

Answer

x^2 + C

Question 26

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

Accepted Answer

(x + 2)e^x + C

Answer

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

Answer

(x - 1)e^x + C

Answer

(x+2)e^x

Question 27

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

Accepted Answer

arctan(x) + C

Answer

1/x + C

Answer

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

Answer

x/2 + C

Question 28

Which of the following is NOT a property of the indefinite integral?

Accepted Answer

The indefinite integral is an exact value.

Answer

The indefinite integral contains an arbitrary constant.

Answer

The indefinite integral of a derivative is the original function.

Question 29

Which of the following integrals represents the area under the curve y = x^3 from x = 0 to x = 2?

Accepted Answer

∫[0, 2] x^3 dx

Answer

∫[0, 2] x^2 dx

Answer

∫[2, 0] x^3 dx

Answer

∫[0, 2] 3x^2 dx

Question 30

Which of the following techniques is used to find the indefinite integral of an expression containing both algebraic and logarithmic terms?

Accepted Answer

Integration by parts

Answer

Trigonometric substitution

Answer

Chain rule

Answer

Partial fractions

Question 31

Find the indefinite integral of the expression √(x^2 - 1).

Accepted Answer

1/2 (x√(x^2 - 1) + arcsin(x)) + C

Answer

x^2 - arcsin(x) + C

Answer

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

Answer

1/2 (x√(x^2 - 1) - arcsin(x)) + C

Question 32

Find the indefinite integral of cos(x).

Accepted Answer

sin(x) + C

Answer

sec²(x) + C

Answer

tan(x) + C

Answer

-sin(x) + C

Question 33

What is the indefinite integral of 1/x?

Accepted Answer

ln|x| + C

Answer

xln(x) - x + C

Answer

-1/x² + C

Answer

eˣ + C

Question 34

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

Accepted Answer

x^2 + C

Answer

2x^2 + C

Answer

x^2

Answer

2x

Question 35

Which of the following is the derivative of  F(x) = sin(x^2)?

Accepted Answer

2x cos(x^2)

Answer

sin(2x)

Answer

cos(x^2)

Answer

2 cos(x^2)

Question 36

What is the indefinite integral of  g(x) = e^(3x) dx?

Accepted Answer

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

Answer

3e^(3x) + C

Answer

e^(3x) + C

Answer

e^(3x)

Question 37

Evaluate the indefinite integral of  h(x) = 1/(x+1) dx.

Accepted Answer

ln|x+1| + C

Answer

1/(x+1)^2 + C

Answer

ln(x+1) + C

Answer

x+1 + C

Question 38

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

Accepted Answer

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

Answer

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

Answer

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

Question 39

Using the substitution method, what is the indefinite integral of  p(x) = x / (x^2 + 1) dx?

Accepted Answer

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

Answer

(1/2)ln(x) + C

Answer

ln(x^2 + 1) + C

Answer

ln(x) + C

Question 40

What is the indefinite integral of  q(x) = cos(x) / sin(x) dx?

Accepted Answer

ln|sin(x)| + C

Answer

sin(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Question 41

If the derivative of  F(x) is equal to  f(x), what can we conclude about  F(x) in relation to  f(x)?

Accepted Answer

F(x) is the indefinite integral of f(x).

Answer

F(x) and f(x) are equal.

Answer

F(x) is the derivative of f(x).

Answer

F(x) and f(x) are unrelated.

Question 42

Given  r(x) = 1/x^2, what is the indefinite integral of  r(x) dx?

Accepted Answer

-1/x + C

Answer

-x + C

Answer

x + C

Answer

1/x + C

Question 43

Which of the following expressions represents a family of functions whose derivative is  s(x) = 2x + 1?

Accepted Answer

x^2 + x + C

Answer

x^2 + C

Answer

2x^2 + x + C

Question 44

What is the indefinite integral of $3x^2 + 2x - 5$ with respect to $x$?

Accepted Answer

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

Answer

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

Answer

$6x + 2 + C$

Answer

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

Question 45

Evaluate the indefinite integral: $\int (2e^x + \frac{1}{x}) dx$

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 46

Which of the following statements about indefinite integrals is TRUE?

Accepted Answer

The indefinite integral of a function is a family of functions.

Answer

The indefinite integral of a function is a unique function.

Answer

The indefinite integral of a function is always a constant.

Question 47

What is the indefinite integral of  `3x^2 + 2x` with respect to `x`?

Accepted Answer

x^3 + x^2 + C

Answer

3x^3 + 2x^2 + C

Answer

x^3 + x + C

Answer

6x + 2 + C

Question 48

Which of the following is the indefinite integral of `cos(x)` with respect to `x`?

Accepted Answer

sin(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C

Answer

tan(x) + C

Question 49

Find the indefinite integral of `e^x` with respect to `x`.

Accepted Answer

e^x + C

Answer

ln(x) + C

Answer

x * e^x + C

Answer

e^(x+1) + C

Question 50

Evaluate the indefinite integral of `(2x + 1)^2` with respect to `x` using substitution.

Accepted Answer

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

Answer

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

Answer

4x^3 + 6x^2 + 3x + C

Answer

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

Question 51

Find the indefinite integral of `x * sin(x)` with respect to `x` using integration by parts.

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 52

What is the indefinite integral of `1/(x+2)` with respect to `x`?

Accepted Answer

ln(|x+2|) + C

Answer

log(x+2) + C

Answer

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

Answer

1/(x+2)^2 + C

Question 53

If `F(x)` is an antiderivative of `f(x)`, which of the following is also an antiderivative of `f(x)`?

Accepted Answer

F(x) + 5

Answer

F(x) * 5

Answer

F(x) - 5

Answer

F(x)/5

Question 54

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

Accepted Answer

tan(x) + C

Answer

cot(x) + C

Answer

sec(x) + C

Answer

-tan(x) + C

Question 55

Evaluate the indefinite integral of `x^2 * e^x` with respect to `x` using integration by parts.

Accepted Answer

x^2 * e^x - 2x * e^x + 2 * e^x + C

Answer

x^2 * e^x - 2x * e^x - 2 * e^x + C

Answer

x^2 * e^x + 2x * e^x + 2 * e^x + C

Answer

x^2 * e^x + 2x * e^x - 2 * e^x + C

Question 56

Which of the following represents the indefinite integral of `sin(2x)` with respect to `x`?

Accepted Answer

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

Answer

2 * cos(2x) + C

Answer

-2 * cos(2x) + C

Answer

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