Further antiderivatives | Year 12

01 / LEARN

Understand the idea.

Reverse differentiation

An antiderivative reverses a derivative: $F^{'} (x) = f (x)$ . For a power $x^{n}$ with $n \neq - 1$ , increase the exponent by one and divide by that new exponent. Integrate sums term by term, retaining each coefficient.

The exception is $\int \frac{1}{x} d x = ln ∣ x ∣ + C$ on an interval excluding zero. For a composite exponential or trigonometric function, compensate for its inner derivative; differentiation should recover the entire original integrand.

Use conditions and domains

The constant $C$ describes the vertical shifts that differentiation cannot distinguish. To use an initial value, integrate first, substitute both coordinates into the resulting family, and solve for $C$ . The initial output is not automatically the constant.

State where the antiderivative is real and differentiable. Reciprocal and logarithmic expressions cannot cross their excluded points; separate intervals may carry different constants. Check the final function against both its derivative and any initial condition.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

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

Worked solution

1
Integrate each power.
$\int (3 x^{2} - 2 x^{- 2}) d x = \frac{3 x^{3}}{3} - \frac{2 x^{- 1}}{- 1} + C$
The power rule divides by the new exponent, including its sign.
2
Simplify and state the domain.
$x^{3} + \frac{2}{x} + C, x \neq 0$
The reciprocal term excludes zero; the formula applies on either interval beside zero.
3
Differentiate to check.
$\frac{d}{d x} (x^{3} + \frac{2}{x} + C) = 3 x^{2} - 2 x^{- 2}$
The derivative reproduces both the coefficient and sign of each original term.

Answer

$x^{3} + 2 / x + C$ , for $x \neq 0$ .

EXAMPLE 02

Worked example 2

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

Worked solution

1
Integrate the given derivative.
$F (x) = \int 4 e^{2 x} d x = 2 e^{2 x} + C$
Dividing by the inner derivative 2 compensates for the chain rule.
2
Apply the initial value.
$3 = 2 e^{0} + C = 2 + C, C = 1$
The condition specifies the value of the whole function at zero.
3
State and verify the function.
$\begin{array}{c} F (x) = 2 e^{2 x} + 1 \\ F^{'} (x) = 4 e^{2 x}, F (0) = 3 \end{array}$
Both the derivative and initial value agree with the requirements.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

It would divide by zero.

For $n = - 1$ , the proposed power-rule denominator is $n + 1 = 0$ . Also, the numerator would be $x^{0}$ , whose derivative is zero. The correct antiderivative is $ln ∣ x ∣ + C$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

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

Read the worked solutions

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

Find $\int x^{\frac{3}{2}} d x$ for $x > 0$ .
2 marks
Increase the exponent to $\frac{5}{2}$ .
Divide by $\frac{5}{2}$ .
$\frac{2}{5} x^{\frac{5}{2}} + C$ .
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 $\frac{1}{3}$ cancels the inner derivative 3.
$\frac{1}{3} sin (3 x) + C$ .
Find $\int (\frac{5}{x} - \frac{2}{x^{3}}) d x$ .
2 marks
Use the logarithm rule for $\frac{1}{x}$ .
Use the power rule for $- 2 x^{- 3}$ .
$5 ln ∣ x ∣ + x^{- 2} + C$ , $x \neq 0$ .
Given $F^{'} = 6 x$ 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 $\frac{1}{\sqrt{x}}$ for $x > 0$ .
2 marks
Differentiate $2 x^{\frac{1}{2}}$ .
The result is $x^{- \frac{1}{2}}$ .
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.

Official curriculum source ↗

In this lesson

Understand the idea.

Reverse differentiation

Use conditions and domains

See the method in action.

Worked example 1

Integrate each power.

Simplify and state the domain.

Differentiate to check.

Worked example 2

Integrate the given derivative.

Apply the initial value.

State and verify the function.

Try it yourself.

Check your understanding

Keep practising