Power/exponential/trigonometric antiderivatives | Year 12

01 / LEARN

The idea, step by step

Match the derivative pattern

The reverse rule must undo the complete derivative. Since the derivative of $e^{3 x}$ is $3 e^{3 x}$ , an antiderivative of $e^{3 x}$ needs a factor of $1 / 3$ . The same adjustment works for sine and cosine of a linear expression.

For trigonometric integrals, angles are in radians. Also keep the sign: differentiating cosine gives negative sine, so integrating sine produces negative cosine.

\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + C (a \neq 0)

Combine rules carefully

Write fractional powers before using the power rule: $\sqrt{x} = x^{1 / 2}$ and $1 / x^{2} = x^{- 2}$ . The power rule does not cover $x^{- 1}$ ; reciprocal linear terms are handled in the next lesson.

Linearity lets you integrate a sum of different function types. Check every coefficient by differentiating the complete result; the derivative check is especially useful for a negative linear input.

\int cos (a x + b) d x = \frac{sin (a x + b)}{a} + C; \int sin (a x + b) d x = - \frac{cos (a x + b)}{a} + C

A common trap

Increasing the power correctly but forgetting to divide by the new exponent or the linear-input coefficient.

Instead: Differentiate the proposed result. Any extra multiplier shows exactly which factor still needs to be divided out.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

Find $\int (5 x^{4} - 3 e^{x} + 2 cos x) d x$ .

Use the power rule on the first term.
The natural exponential is unchanged on integration.
The antiderivative of cosine is sine; retain coefficient 2.

x^{5} - 3 e^{x} + 2 sin x + C

Example 2

Worked example

Find $\int cos (3 x) d x$ .

The derivative of $sin (3 x)$ is $3 cos (3 x)$ .
Divide by 3 to undo that factor.
Differentiate the answer to recover the integrand.

sin (3 x) / 3 + C

Check your understanding

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

Show answer

$e^{2 x} + C$

The derivative of the answer is $2 e^{2 x}$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find $\int (8 x^{3} + e^{x}) d x$ .
3 marks
Find $\int \sqrt{x} d x$ for $x > 0$ .
3 marks
Find $\int x^{- 2} d x$ on $x > 0$ .
3 marks
Find $\int 4 e^{- 2 x} d x$ .
3 marks
Find $\int 3 sin (2 x) d x$ .
3 marks
Find $\int (2 cos x - sin x) d x$ .
3 marks

Read the worked solutions

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

Find $\int (8 x^{3} + e^{x}) d x$ .
3 marks
Apply the power and natural-exponential rules.
$2 x^{4} + e^{x} + C$
Find $\int \sqrt{x} d x$ for $x > 0$ .
3 marks
Rewrite as $x^{1 / 2}$ , then divide $x^{3 / 2}$ by $3 / 2$ .
$\frac{2}{3} x^{3 / 2} + C$
Find $\int x^{- 2} d x$ on $x > 0$ .
3 marks
Raise the power to $- 1$ and divide by $- 1$ .
$- x^{- 1} + C$
Find $\int 4 e^{- 2 x} d x$ .
3 marks
Divide coefficient 4 by $- 2$ .
$- 2 e^{- 2 x} + C$
Find $\int 3 sin (2 x) d x$ .
3 marks
Use the negative-cosine antiderivative and divide by 2.
$- \frac{3}{2} cos (2 x) + C$
Find $\int (2 cos x - sin x) d x$ .
3 marks
Integrate the two terms separately; check both signs.
$2 sin x + cos x + C$

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Antiderivatives involving reciprocal linear expressions, linked back to logarithmic derivatives

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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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