Antiderivatives and constants | Year 12

01 / LEARN

The idea, step by step

A family, not a single answer

An antiderivative of a function is another function whose derivative gives the original function. For example, both $x^{2} + 3$ and $x^{2} - 7$ differentiate to $2 x$ . Differentiation loses a constant, so reversing it cannot identify that constant without extra information.

An indefinite integral describes every antiderivative on the interval being considered. Write one suitable expression and add $C$ , where $C$ is any real constant.

\int 2 x d x = x^{2} + C

Reverse each term and check

For a power of $x$ , raise the power by one and divide by the new power. Constants integrate to a multiple of $x$ . Use linearity to handle a sum one term at a time.

A fast and reliable check is to differentiate your proposed answer. You must recover the whole original integrand, including every coefficient and sign. This check also distinguishes a harmless different-looking antiderivative from an incorrect one.

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C (n \neq - 1)

A common trap

Writing only one antiderivative as though the constant is known.

Instead: Use $+ C$ for an indefinite integral. A supplied point or initial value is needed to determine it.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

Find $\int (6 x^{2} - 4) d x$ .

Integrate $6 x^{2}$ : $6 x^{3} / 3 = 2 x^{3}$ .
Integrate $- 4$ : $- 4 x$ .
Include one arbitrary constant.
Differentiate to check: $6 x^{2} - 4$ .

2 x^{3} - 4 x + C

Example 2

Worked example

Are $F (x) = x^{2} + 5$ and $G (x) = x^{2} - 2$ both antiderivatives of $2 x$ ?

$F^{'} (x) = 2 x$ and $G^{'} (x) = 2 x$ .
Their difference is the constant 7, which disappears on differentiation.
Both belong to $x^{2} + C$ .

Yes.

Check your understanding

Find $\int 3 x^{2} d x$ .

Show answer

$x^{3} + C$

Differentiating $x^{3} + C$ gives $3 x^{2}$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $\int 4 x d x$ .
3 marks
Find $\int (9 x^{2} + 2) d x$ .
3 marks
Find $\int (x^{3} - 5 x) d x$ .
3 marks
Find $\int 7 d x$ and explain the variable in the answer.
3 marks
A student claims $3 x^{2}$ is an antiderivative of $3 x$ . Check and correct the claim.
3 marks
For $f (x) = 2 x + 1$ , give two different antiderivatives and explain their relationship.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $\int 4 x d x$ .
3 marks
Raise the power to 2, then divide $4 x^{2}$ by 2.
Add $C$ .
$2 x^{2} + C$
Find $\int (9 x^{2} + 2) d x$ .
3 marks
$9 x^{2}$ integrates to $3 x^{3}$ ; $2$ integrates to $2 x$ .
$3 x^{3} + 2 x + C$
Find $\int (x^{3} - 5 x) d x$ .
3 marks
Integrate each power separately and add $C$ .
$x^{4} / 4 - 5 x^{2} / 2 + C$
Find $\int 7 d x$ and explain the variable in the answer.
3 marks
The derivative of $7 x$ is 7; the integral is with respect to $x$ .
$7 x + C$
A student claims $3 x^{2}$ is an antiderivative of $3 x$ . Check and correct the claim.
3 marks
The derivative of $3 x^{2}$ is $6 x$ , not $3 x$ .
Divide by 2 before differentiating back.
Incorrect; $3 x^{2} / 2 + C$ is the family.
For $f (x) = 2 x + 1$ , give two different antiderivatives and explain their relationship.
3 marks
Both differentiate to $2 x + 1$ .
They differ by a constant.
For example $x^{2} + x$ and $x^{2} + x + 4$ .

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Power/exponential/trigonometric antiderivatives

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Mathematical Methods integration and logarithmic antiderivatives; original learning and assessment material, with by-hand methods and explicitly described numerical models.

Official curriculum source ↗

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