Further antiderivatives | Year 12

01 / LEARN

The idea, step by step

Reverse differentiation

An antiderivative F of f satisfies $F^{'} (x) = f (x)$ on a stated interval. For $n \neq - 1$ , $\int x^{n} d x = x^{n + 1} / (n + 1) + C$ . The exception is important: $\int 1 / x d x = ln ∣ x ∣ + C$ on intervals not crossing zero.

Integrate a sum term by term and keep constant factors. Chain factors still matter: $\int e^{a x + b} d x = e^{a x + b} / a + C$ for nonzero a. Differentiate a proposed answer to check both its form and its coefficient.

Use conditions and domains

The integration constant labels an entire family. A condition such as F(0)=3 determines one member after integration. Do not replace C with the initial value before evaluating the other terms at that input.

Fractional powers, logarithms and reciprocal expressions bring domain restrictions. State an interval where the answer is real and differentiable. Different constants can be needed on disconnected intervals; a graphing calculator cannot remove that mathematical restriction.

A common trap

Increase the power but forget to divide by the new power.

Instead: Differentiate your answer: the coefficient must return to the original integrand.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

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

Apply the power rule to each term.
Differentiate: $3 x^{2} - 2 / x^{2}$ , matching the integrand.

x^{3} + 2 / x + C

, for

x \neq 0

Example 2

Worked example 2

Given $F^{'} (x) = 4 e^{2 x}$ and F(0)=3, find F.

Integrate to $2 e^{2 x} + C$ .
At x=0, $2 + C = 3$ , so C=1.

F (x) = 2 e^{2 x} + 1

Check your understanding

Why does the power rule not work for n=-1?

Show answer

It would divide by zero.

The antiderivative of 1/x is logarithmic, not a power.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $\int x^{3 / 2} d x$ for x>0.
2 marks
Find $\int (2 e^{3 x} + 4) d x$ .
2 marks
Find $\int cos (3 x) d x$ .
2 marks
Find $\int (5 / x - 2 / x^{3}) d x$ .
2 marks
Given F'=6x and F(1)=5, find F.
2 marks
Check whether $2 \sqrt{x} + C$ is an antiderivative of $1 / \sqrt{x}$ for x>0.
2 marks

Read the worked solutions

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

Find $\int x^{3 / 2} d x$ for x>0.
2 marks
Increase the exponent to 5/2.
Divide by 5/2.
$\frac{2}{5} x^{5 / 2} + C$ .
Find $\int (2 e^{3 x} + 4) d x$ .
2 marks
Divide the exponential coefficient by 3.
Integrate 4 as 4x.
$\frac{2}{3} e^{3 x} + 4 x + C$ .
Find $\int cos (3 x) d x$ .
2 marks
Sine differentiates to cosine.
The factor 1/3 cancels the inner derivative 3.
$\frac{1}{3} sin (3 x) + C$ .
Find $\int (5 / x - 2 / x^{3}) d x$ .
2 marks
Use the logarithm rule for 1/x.
Use the power rule for $- 2 x^{- 3}$ .
$5 ln ∣ x ∣ + x^{- 2} + C$ , x not zero.
Given F'=6x and F(1)=5, find F.
2 marks
Integrate to $3 x^{2} + C$ .
Use $3 + C = 5$ .
$F = 3 x^{2} + 2$ .
Check whether $2 \sqrt{x} + C$ is an antiderivative of $1 / \sqrt{x}$ for x>0.
2 marks
Differentiate $2 x^{1 / 2}$ .
The result is $x^{- 1 / 2}$ .
Yes.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics integration techniques and applications, including inverse-trigonometric patterns, substitution, parts, simple partial fractions, areas and solids of revolution.

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