Substitution | Year 12 | Ascent Library

01 / LEARN

The idea, step by step

Reverse a chain rule

Substitution works when an inner expression and a multiple of its derivative occur together. Let u equal the inner expression, calculate du, and replace every remaining x term before integrating. The substitution is an equality of differentials, not permission to ignore a coefficient.

For $\int x {(1 + x^{2})}^{4} d x$ , u=1+x squared gives du=2x dx, so x dx=du/2. The transformed integral is a power of u. A derivative check in x confirms the factor.

Keep variables and limits consistent

For a definite integral, change the endpoints into u values and finish in u. Alternatively, return to x before using the original endpoints. Do not use x endpoints in an antiderivative still written in u.

A helpful substitution must simplify the whole integrand. If an unexplained x remains after substitution, either express it in u using the domain information or choose another method. State any restrictions needed for the substitution.

A common trap

Change the variable but keep the original endpoints without converting back.

Instead: Limits and antiderivative must refer to the same variable.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

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

Set u=1+x squared, du=2x dx.
Integrate $\frac{1}{2} u^{4}$ and substitute back.

{(1 + x^{2})}^{5} / 10 + C

Example 2

Worked example 2

Evaluate $\int_{0}^{1} 2 x e^{x^{2}} d x$ .

Use u=x squared. The new limits are 0 and 1.
Integrate $e^{u}$ between those limits.

e - 1

Check your understanding

For u=3x+1, what is dx?

Show answer

$d u / 3$ .

Since du=3dx, divide by 3.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find $\int 2 x / (x^{2} + 1) d x$ .
2 marks
Find $\int cos x {sin}^{3} x d x$ .
2 marks
Find $\int x / \sqrt{4 + x^{2}} d x$ .
2 marks
Find $\int {(3 x + 1)}^{5} d x$ .
2 marks
Find $\int e^{2 x} / (1 + e^{2 x}) d x$ .
2 marks
Evaluate $\int_{0}^{2} x (1 + x^{2}) d x$ .
2 marks

Read the worked solutions

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

Find $\int 2 x / (x^{2} + 1) d x$ .
2 marks
Use u=x squared+1.
The integral becomes $\int d u / u$ ; u is positive.
$ln (x^{2} + 1) + C$ .
Find $\int cos x {sin}^{3} x d x$ .
2 marks
Set u=sin x and du=cos x dx.
Integrate u cubed.
${sin}^{4} x / 4 + C$ .
Find $\int x / \sqrt{4 + x^{2}} d x$ .
2 marks
Use u=4+x squared and x dx=du/2.
Integrate $u^{- 1 / 2} / 2$ .
$\sqrt{4 + x^{2}} + C$ .
Find $\int {(3 x + 1)}^{5} d x$ .
2 marks
Use u=3x+1, dx=du/3.
Integrate u to the fifth and divide by 3.
${(3 x + 1)}^{6} / 18 + C$ .
Find $\int e^{2 x} / (1 + e^{2 x}) d x$ .
2 marks
Use u=1+exp(2x); du=2exp(2x)dx.
Integrate du/(2u).
$\frac{1}{2} ln (1 + e^{2 x}) + C$ .
Evaluate $\int_{0}^{2} x (1 + x^{2}) d x$ .
2 marks
With u=1+x squared, the limits become 1 and 5.
$\frac{1}{2} \int_{1}^{5} u d u = (25 - 1) / 4 = 6$ .
$6$ .

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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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