Inverse-trigonometric derivatives and integrals | Year 12

01 / LEARN

The idea, step by step

Recognise the two basic patterns

For a>0, $\int 1 / \sqrt{a^{2} - x^{2}} d x = arcsin (x / a) + C$ . Its real differentiable domain is inside -a<x<a. Also $\int 1 / (a^{2} + x^{2}) d x = (1 / a) arctan (x / a) + C$ , valid for all real x.

These formulas use radians. The two denominators have different shapes: a difference under a square root suggests arcsine, whereas a sum of squares without a root suggests arctangent. First factor or complete a square if the pattern is hidden.

Check scaling with a derivative

A scaled input brings an inner derivative. For example, the derivative of arctan(2x) is $2 / (1 + 4 x^{2})$ , not $1 / (1 + 4 x^{2})$ . Introduce a substitution if the scaling is not immediately clear.

An inverse trigonometric function is not a reciprocal. $arcsin x$ means the inverse sine value on its principal range, while $1 / sin x$ is cosecant. Correct notation and a derivative check prevent this common confusion.

A common trap

Write arcsin x as 1/sin x.

Instead: Inverse-function notation and reciprocal notation describe different functions.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Find $\int 1 / \sqrt{9 - x^{2}} d x$ .

Compare with the arcsine pattern using a=3.
Differentiate the answer to recover the denominator.

arcsin (x / 3) + C

, -3<x<3.

Example 2

Worked example 2

Find $\int 1 / (x^{2} + 4) d x$ .

Here a=2.
The outside factor 1/2 is essential.

\frac{1}{2} arctan (x / 2) + C

Check your understanding

Differentiate arctan(3x).

Show answer

$3 / (1 + 9 x^{2})$ .

Apply the inner derivative 3.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find $\int 3 / (9 + x^{2}) d x$ .
2 marks
Find $\int 2 / \sqrt{1 - 4 x^{2}} d x$ .
2 marks
Find $\int 1 / (1 + 9 x^{2}) d x$ .
2 marks
Evaluate $\int_{0}^{1 / 2} 1 / \sqrt{1 - x^{2}} d x$ .
2 marks
Find $\int 1 / (x^{2} + 16) d x$ .
2 marks
Differentiate $arccos (x / 2)$ on -2<x<2.
2 marks

Read the worked solutions

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

Find $\int 3 / (9 + x^{2}) d x$ .
2 marks
Use the arctangent rule with a=3.
The factor 3 cancels the outside 1/3.
$arctan (x / 3) + C$ .
Find $\int 2 / \sqrt{1 - 4 x^{2}} d x$ .
2 marks
Use u=2x and du=2dx.
The transformed integral is $\int {(1 - u^{2})}^{- 1 / 2} d u$ .
$arcsin (2 x) + C$ , $∣ x ∣ < 1 / 2$ .
Find $\int 1 / (1 + 9 x^{2}) d x$ .
2 marks
Differentiate arctan(3x) to see the extra factor 3.
Divide by 3.
$\frac{1}{3} arctan (3 x) + C$ .
Evaluate $\int_{0}^{1 / 2} 1 / \sqrt{1 - x^{2}} d x$ .
2 marks
The antiderivative is arcsin x.
Evaluate arcsin(1/2)-arcsin(0).
$π / 6$ .
Find $\int 1 / (x^{2} + 16) d x$ .
2 marks
Use a=4 in the arctangent pattern.
Check the chain factor.
$\frac{1}{4} arctan (x / 4) + C$ .
Differentiate $arccos (x / 2)$ on -2<x<2.
2 marks
Use derivative $- u^{'} / \sqrt{1 - u^{2}}$ .
Simplify with u=x/2.
$- 1 / \sqrt{4 - x^{2}}$ .

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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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