Antiderivatives and constants | Year 12

01 / LEARN

Understand the idea.

A family, not a single answer

An antiderivative reverses a derivative: if $F^{'} (x) = f (x)$ , then $F$ is an antiderivative of $f$ . Adding a constant changes the height of the graph but leaves every gradient unchanged.

The symbol $\int f (x) d x$ asks for the whole family $F (x) + C$ on the interval considered. For example, $x^{2} + 3$ and $x^{2} - 7$ both differentiate to $2 x$ ; the derivative cannot recover their different constants.

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

Reverse each term and check

To reverse a power derivative, increase the exponent by one and divide by that new exponent. The rule requires $n \neq - 1$ . Integrate sums term by term, retaining the coefficient and sign of each term.

Check the result by differentiating it. Recovering every term of the integrand verifies the calculation; it does not determine $C$ . A constant integrand needs an $x$ because differentiating a multiple of $x$ returns that constant.

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

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

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

Worked solution

1
Integrate the quadratic term.
$\int 6 x^{2} d x = 2 x^{3} + C_{1}$
Increasing the power to 3 introduces a factor of 3 on differentiation, so divide the coefficient 6 by 3.
2
Integrate the constant term.
$\int (- 4) d x = - 4 x + C_{2}$
A constant rate accumulates in direct proportion to the input.
3
Combine the terms and include the family.
$\int (6 x^{2} - 4) d x = 2 x^{3} - 4 x + C$
Linearity allows separate integration, and one arbitrary constant represents the combined family.
4
Differentiate to verify.
$\frac{d}{d x} (2 x^{3} - 4 x + C) = 6 x^{2} - 4$
The derivative recovers the complete integrand.

Answer

$2 x^{3} - 4 x + C$

EXAMPLE 02

Worked example

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

Worked solution

1
Differentiate the first function.
$F^{'} (x) = \frac{d}{d x} (x^{2} + 5) = 2 x$
The derivative of the added constant is zero.
2
Differentiate the second function.
$G^{'} (x) = \frac{d}{d x} (x^{2} - 2) = 2 x$
The different constant also disappears on differentiation.
3
Identify their relationship.
$F (x) - G (x) = 7$
Both are antiderivatives of the specified function; they are members of the family with different constant values.

Answer

Yes.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$x^{3} + C$

The power rule gives $\int 3 x^{2} d x = \frac{3 x^{3}}{3} + C = x^{3} + C$ . Differentiating verifies $3 x^{2}$ ; the constant remains undetermined.

Keep practising

6 worksheet questions · 18 marks

Open worksheet 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 $C$ .
$\frac{x^{4}}{4} - \frac{5 x^{2}}{2} + C$
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; $\frac{3 x^{2}}{2} + C$ 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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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 ↗

In this lesson

Understand the idea.

A family, not a single answer

Reverse each term and check

See the method in action.

Worked example

Integrate the quadratic term.

Integrate the constant term.

Combine the terms and include the family.

Differentiate to verify.

Worked example

Differentiate the first function.

Differentiate the second function.

Identify their relationship.

Try it yourself.

Check your understanding

Keep practising