Indefinite Integral — Basic

Indefinite Integral — Basic. Practice questions to deepen understanding of the indefinite integral — basic level. Online math practice with full solutions and step-by-step explanations.

Indefinite Integral — Basic. The integral as the inverse operation of differentiation, the constant of integration, basic formulas. Explanations for beginners.

30 questions

Question 1

3.33 pts

What is an indefinite integral?

The reverse of differentiation — finding a function whose derivative is the given function

Computing the area under the graph

Finding extreme points

Computing the derivative of a function

Explanation:

Explanation: Integration (anti-differentiation) reverses differentiation.

Question 2

3.33 pts

Why do we add +C in an indefinite integral?

Because infinitely many functions share the same derivative, differing only by a constant

Because it is a required mathematical convention

To make the answer more accurate

Because C represents the area

Explanation:

Explanation: Any constant has derivative 0, so F(x)+C and F(x) have the same derivative.

Question 3

3.33 pts

What happens when we differentiate \(\int f(x)\,dx\)?

We get back the original function f(x)

We get zero

We get the constant C

We get a double integral

Explanation:

Explanation: Differentiation and integration are inverse operations.

Question 4

3.33 pts

What is the rule for \(\int x^n\,dx\)?

\(\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C\)

\(\int x^n\,dx = nx^{n-1} + C\)

\(\int x^n\,dx = x^{n+1} + C\)

\(\int x^n\,dx = \dfrac{x^n}{n} + C\)

Explanation:

Explanation: Power rule for integration: raise exponent by 1, divide by new exponent.

Question 5

3.33 pts

What is \(\int 5\,dx\)?

5x + C

5 + C

x + C

Explanation:

Explanation: Integral of a constant k: kx + C.

Question 6

3.33 pts

What is \(\int (f(x)+g(x))\,dx\)?

\(\int f(x)\,dx + \int g(x)\,dx\)

\(\int f(x)\,dx \cdot \int g(x)\,dx\)

\(\int f(x)\cdot g(x)\,dx\)

Cannot be computed

Explanation:

Explanation: Linearity of integration: integral of a sum = sum of integrals.

Question 7

3.33 pts

What is \(\int c\,f(x)\,dx\) where c is a constant?

\(c\int f(x)\,dx\)

\(\int f(cx)\,dx\)

\(c + \int f(x)\,dx\)

\(\dfrac{1}{c}\int f(x)\,dx\)

Explanation:

Explanation: Constants factor out of integrals.

Question 8

3.33 pts

Why is integration called the "inverse operation" of differentiation?

Because integration undoes differentiation

Because they use the same rules

Because both compute areas

Because both require limits

Explanation:

Explanation: d/dx[∫f dx] = f(x).

Question 9

3.33 pts

What is the difference between \(\int x^2\,dx\) and \(\int_0^1 x^2\,dx\)?

The first is indefinite (a function); the second is definite (a number)

The first is easier to compute

The first has +C; the second does not

There is no meaningful difference

Explanation:

Explanation: Indefinite integral = family of functions; definite integral = specific number.

Question 10

3.33 pts

How do you verify an indefinite integral?

Differentiate the result and check that you recover the original function

Substitute x=0 and check

Compute the area under the graph

Compare with C

Explanation:

Explanation: d/dx[F(x)+C] = f(x) confirms correctness.

Question 11

3.33 pts

What is \(\int 0\,dx\)?

C (constant only)

Undefined

Explanation:

Explanation: The anti-derivative of 0 is any constant C.

Question 12

3.33 pts

What is \(\int (2x+3x)\,dx\)?

\(\int 5x\,dx = \dfrac{5x^2}{2} + C\)

Must compute separately

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

Cannot simplify

Explanation:

Explanation: Simplify first: 2x+3x=5x. Then ∫5x dx=5x²/2+C.

Question 13

3.33 pts

What is \(\int x^0\,dx\)?

x + C

Explanation:

Explanation: \(x^0=1\), so \(\int 1\,dx=x+C\).

Question 14

3.33 pts

Which differentiation rule becomes the integration rule?

The power rule

The product rule

The chain rule

The quotient rule

Explanation:

Explanation: Integration reverses the power rule: ∫xⁿdx = xⁿ⁺¹/(n+1)+C.

Question 15

3.33 pts

Why does an indefinite integral have no limits?

Because we seek a general function, not a numerical value

Because it is easier

Because we do not know the limits

Because they are not needed

Explanation:

Explanation: Indefinite integral = anti-derivative family; limits would give a specific number.

Question 16

3.33 pts

Compute \(\int x\,dx\).

\(\dfrac{x^2}{2}+C\)

\(x^2+C\)

\(\dfrac{x}{2}+C\)

\(2x+C\)

Explanation:

Explanation: Power rule: \(\int x\,dx=\dfrac{x^2}{2}+C\).

Question 17

3.33 pts

Compute \(\int x^2\,dx\).

\(\dfrac{x^3}{3}+C\)

\(x^3+C\)

\(2x+C\)

\(\dfrac{x^2}{2}+C\)

Explanation:

Explanation: \(\int x^2\,dx=\dfrac{x^3}{3}+C\).

Question 18

3.33 pts

Compute \(\int x^3\,dx\).

\(\dfrac{x^4}{4}+C\)

\(x^4+C\)

\(3x^2+C\)

\(\dfrac{x^3}{3}+C\)

Explanation:

Explanation: \(\int x^3\,dx=\dfrac{x^4}{4}+C\).

Question 19

3.33 pts

Compute \(\int 3\,dx\).

3x + C

3 + C

x + C

\(\dfrac{3x^2}{2}+C\)

Explanation:

Explanation: \(\int k\,dx=kx+C\). So 3x+C.

Question 20

3.33 pts

Compute \(\int 2x\,dx\).

\(x^2+C\)

\(2x^2+C\)

\(\dfrac{x^2}{2}+C\)

\(2x+C\)

Explanation:

Explanation: \(\int 2x\,dx=2\cdot\dfrac{x^2}{2}+C=x^2+C\).

Question 21

3.33 pts

Compute \(\int 5x^2\,dx\).

\(\dfrac{5x^3}{3}+C\)

\(5x^3+C\)

\(\dfrac{x^3}{3}+C\)

\(10x+C\)

Explanation:

Explanation: \(5\cdot\dfrac{x^3}{3}+C=\dfrac{5x^3}{3}+C\).

Question 22

3.33 pts

Compute \(\int (x+3)\,dx\).

\(\dfrac{x^2}{2}+3x+C\)

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

\(\dfrac{x^2}{2}+C\)

\(x+3x+C\)

Explanation:

Explanation: \(\dfrac{x^2}{2}+3x+C\).

Question 23

3.33 pts

Compute \(\int (2x+5)\,dx\).

\(x^2+5x+C\)

\(2x^2+5x+C\)

\(\dfrac{x^2}{2}+5x+C\)

\(2x+5+C\)

Explanation:

Explanation: \(x^2+5x+C\).

Question 24

3.33 pts

Compute \(\int (x^2+x)\,dx\).

\(\dfrac{x^3}{3}+\dfrac{x^2}{2}+C\)

\(\dfrac{x^3}{3}+x+C\)

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

\(\dfrac{x^3+x^2}{2}+C\)

Explanation:

Explanation: \(\dfrac{x^3}{3}+\dfrac{x^2}{2}+C\).

Question 25

3.33 pts

Compute \(\int 4x^3\,dx\).

\(x^4+C\)

\(4x^4+C\)

\(\dfrac{x^4}{4}+C\)

\(12x^2+C\)

Explanation:

Explanation: \(4\cdot\dfrac{x^4}{4}+C=x^4+C\).

Question 26

3.33 pts

Compute \(\int (3x^2-2x)\,dx\).

\(x^3-x^2+C\)

\(3x^3-2x^2+C\)

\(\dfrac{x^3}{3}-\dfrac{x^2}{2}+C\)

\(x^3-2x+C\)

Explanation:

Explanation: \(x^3-x^2+C\).

Question 27

3.33 pts

Compute \(\int (x^2+3x+2)\,dx\).

\(\dfrac{x^3}{3}+\dfrac{3x^2}{2}+2x+C\)

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

\(\dfrac{x^3}{3}+3x+2+C\)

\(\dfrac{x^3+3x^2+2x}{3}+C\)

Explanation:

Explanation: \(\dfrac{x^3}{3}+\dfrac{3x^2}{2}+2x+C\).

Question 28

3.33 pts

Compute \(\int 6x\,dx\).

\(3x^2+C\)

\(6x^2+C\)

\(\dfrac{x^2}{2}+C\)

\(6x+C\)

Explanation:

Explanation: \(6\cdot\dfrac{x^2}{2}+C=3x^2+C\).

Question 29

3.33 pts

Compute \(\int (x^3+x^2+x+1)\,dx\).

\(\dfrac{x^4}{4}+\dfrac{x^3}{3}+\dfrac{x^2}{2}+x+C\)

\(x^4+x^3+x^2+x+C\)

\(\dfrac{x^4+x^3+x^2+x}{4}+C\)

\(\dfrac{x^4}{4}+\dfrac{x^3}{3}+\dfrac{x^2}{2}+C\)

Explanation:

Explanation: Each term integrated separately.

Question 30

3.33 pts

Compute \(\int (5x^4-3x^2+7)\,dx\).

\(x^5-x^3+7x+C\)

\(\dfrac{5x^5}{5}-\dfrac{3x^3}{3}+7x+C\)

\(5x^5-3x^3+7x+C\)

\(\dfrac{x^5}{5}-\dfrac{x^3}{3}+7x+C\)

Explanation:

Explanation: \(5\cdot\dfrac{x^5}{5}-3\cdot\dfrac{x^3}{3}+7x+C=x^5-x^3+7x+C\).