Substitution | Year 12 | Ascent Library

01 / LEARN

Understand the idea.

Reverse a chain rule

Substitution reverses the chain rule. Look for an inner expression together with a constant multiple of its derivative. Define $u$ , differentiate it, and solve for the whole differential factor present in the integral before replacing anything.

For $u = 1 + x^{2}$ , the differential is $d u = 2 x d x$ , so $x d x = \frac{1}{2} d u$ . The factor one half is essential. After substitution, every part of the integral must be expressed in $u$ .

Keep variables and limits consistent

For a definite integral, convert each endpoint using the same definition of $u$ . Evaluate the resulting integral entirely in $u$ . Alternatively, substitute back into $x$ before applying the original bounds; never mix one variable's bounds with another variable's formula.

Choose a substitution that simplifies the whole integrand. If an unwanted $x$ remains, rewrite it using the substitution and relevant domain, or reconsider the choice. For an indefinite integral, return to the original variable and include $C$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

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

Worked solution

1
Choose the inner expression.
$u = 1 + x^{2}, d u = 2 x d x, x d x = \frac{1}{2} d u$
The factor x in the integrand is proportional to the derivative of the bracket.
2
Integrate in the new variable.
$\int x {(1 + x^{2})}^{4} d x = \frac{1}{2} \int u^{4} d u = \frac{u^{5}}{10} + C$
Substitution leaves an ordinary power of u.
3
Return to the original variable.
$\frac{{(1 + x^{2})}^{5}}{10} + C$
The question asks for an antiderivative in x.
4
Check the chain factors.
$\frac{d}{d x} \frac{{(1 + x^{2})}^{5}}{10} = \frac{5}{10} {(1 + x^{2})}^{4} (2 x) = x {(1 + x^{2})}^{4}$
Differentiation confirms that the factor one tenth is necessary.

Answer

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

EXAMPLE 02

Worked example 2

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

Worked solution

1
Substitute for the exponent.
$u = x^{2}, d u = 2 x d x$
The entire factor multiplying the exponential is the differential of its exponent.
2
Convert both bounds.
$x = 0 \Rightarrow u = 0, x = 1 \Rightarrow u = 1$
Bounds must refer to the new integration variable even when their numerical values remain unchanged.
3
Evaluate the exponential integral.
$\int_{0}^{1} 2 x e^{x^{2}} d x = \int_{0}^{1} e^{u} d u = {[e^{u}]}_{0}^{1} = e - 1$
The derivative of the exponential is unchanged by integration in u.

Answer

$e - 1$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$d u / 3$ .

Differentiating $u = 3 x + 1$ gives $d u = 3 d x$ . Dividing by 3 gives $d x = \frac{1}{3} d u$ ; this factor must accompany the transformed integral.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

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

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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.

Official curriculum source ↗

In this lesson

Understand the idea.

Reverse a chain rule

Keep variables and limits consistent

See the method in action.

Worked example 1

Choose the inner expression.

Integrate in the new variable.

Return to the original variable.

Check the chain factors.

Worked example 2

Substitute for the exponent.

Convert both bounds.

Evaluate the exponential integral.

Try it yourself.

Check your understanding

Keep practising