Power/exponential/trigonometric antiderivatives | Year 12

01 / LEARN

Understand the idea.

Match the derivative pattern

Choose an antiderivative whose derivative matches the whole integrand. For a linear input $a x + b$ , differentiation introduces the constant multiplier $a$ . Integration must divide by $a$ , provided $a \neq 0$ .

For trigonometric functions, use radians and check signs: cosine differentiates to negative sine, so the antiderivative of sine is negative cosine. The antiderivative of cosine is sine. Exponential functions keep the same input.

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

Combine rules carefully

Rewrite roots and reciprocals as powers before choosing a rule: $\sqrt{x} = x^{1 / 2}$ and $\frac{1}{x^{2}} = x^{- 2}$ . Apply each rule only on an interval where the function is defined; the exponent $- 1$ requires the logarithmic rule.

A sum can contain several function types. Integrate its terms independently and add one $C$ . Differentiate the entire answer to check every coefficient, especially when the linear input has a negative slope.

\begin{aligned} \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 \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

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

Worked solution

1
Integrate the power term.
$\int 5 x^{4} d x = x^{5} + C_{1}$
Dividing coefficient 5 by the new exponent 5 reverses the power rule.
2
Integrate the exponential term.
$\int (- 3 e^{x}) d x = - 3 e^{x} + C_{2}$
The derivative of the natural exponential is itself.
3
Integrate the cosine term and combine.
$\int (5 x^{4} - 3 e^{x} + 2 cos x) d x = x^{5} - 3 e^{x} + 2 sin x + C$
Sine differentiates to cosine, and the original coefficient 2 is retained.
4
Check by differentiation.
$\frac{d}{d x} (x^{5} - 3 e^{x} + 2 sin x + C) = 5 x^{4} - 3 e^{x} + 2 cos x$
All three original terms and signs are recovered.

Answer

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

EXAMPLE 02

Worked example

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

Worked solution

1
Find the derivative pattern.
$\frac{d}{d x} sin (3 x) = 3 cos (3 x)$
The chain rule contributes the multiplier 3.
2
Undo the extra multiplier.
$\int cos (3 x) d x = \frac{1}{3} sin (3 x) + C$
Dividing by 3 leaves the required cosine coefficient 1.
3
Verify the coefficient.
$\frac{d}{d x} (\frac{sin (3 x)}{3} + C) = \frac{1}{3} \cdot 3 cos (3 x) = cos (3 x)$
The integration adjustment cancels the chain-rule multiplier.

Answer

$sin (3 x) / 3 + C$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$e^{2 x} + C$

Because $\frac{d}{d x} e^{2 x} = 2 e^{2 x}$ , the coefficient already matches the integrand: $\int 2 e^{2 x} d x = e^{2 x} + C$ .

Keep practising

6 worksheet questions · 18 marks

Open worksheet 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 $x^{\frac{1}{2}}$ , then divide $x^{\frac{3}{2}}$ by $\frac{3}{2}$ .
$\frac{2}{3} x^{\frac{3}{2}} + C$
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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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.

Match the derivative pattern

Combine rules carefully

See the method in action.

Worked example

Integrate the power term.

Integrate the exponential term.

Integrate the cosine term and combine.

Check by differentiation.

Worked example

Find the derivative pattern.

Undo the extra multiplier.

Verify the coefficient.

Try it yourself.

Check your understanding

Keep practising