Selecting/combining methods | Year 12

01 / LEARN

The idea, step by step

Ask what differentiation would create

Before calculating, identify a derivative pattern: an inner expression with its derivative suggests substitution; a product that simplifies under differentiation suggests parts; a rational expression may need division or partial fractions.

A square-completion step can reveal an inverse-trigonometric pattern. For example, $x^{2} + 6 x + 13 = {(x + 3)}^{2} + 4$ . Algebraic preparation is part of integration, not a different problem.

Explain why a method fits

Different correct methods can give antiderivatives that differ by a constant. A derivative check tests equivalence more reliably than comparing their appearance. For definite integrals, also check the integrand's sign and the rough size of the answer.

Do not force integration by parts onto every product. In x exp(x squared), the x is already proportional to the derivative of the inner exponent, so substitution is shorter. Combining methods is useful only when each stage actually simplifies the remaining task.

A common trap

Every product requires integration by parts.

Instead: First check for a reverse chain-rule pattern, which may be simpler.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Choose a method and find $\int x e^{x^{2}} d x$ .

The factor x matches the derivative of x squared up to a constant.
Use u=x squared, rather than parts.

e^{x^{2}} / 2 + C

Example 2

Worked example 2

Integrate $1 / (x^{2} + 6 x + 13)$ .

Complete the square: denominator=(x+3) squared+4.
Use the scaled arctangent rule.

\frac{1}{2} arctan ((x + 3) / 2) + C

Check your understanding

Which is shorter for integral ln x / x?

Show answer

Substitution u=ln x.

The factor dx/x is exactly du.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Integrate $ln x / x$ for x>0.
2 marks
Integrate $x / \sqrt{1 - x^{2}}$ on -1<x<1.
2 marks
Integrate $x e^{2 x}$ .
2 marks
Integrate ${cos}^{2} x sin x$ .
2 marks
Explain whether x exp(x squared) and x exp(x) should use the same first method.
2 marks
Verify that $e^{x} (x^{2} - 2 x + 2)$ integrates $x^{2} e^{x}$ .
2 marks

Read the worked solutions

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

Integrate $ln x / x$ for x>0.
2 marks
Set u=ln x.
Integrate u du.
${(ln x)}^{2} / 2 + C$ .
Integrate $x / \sqrt{1 - x^{2}}$ on -1<x<1.
2 marks
Use u=1-x squared.
Then x dx=-du/2.
$- \sqrt{1 - x^{2}} + C$ .
Integrate $x e^{2 x}$ .
2 marks
Use parts with u=x and v=exp(2x)/2.
Subtract the remaining exponential integral.
$e^{2 x} (x / 2 - 1 / 4) + C$ .
Integrate ${cos}^{2} x sin x$ .
2 marks
Use u=cos x.
The negative differential changes the sign.
$- {cos}^{3} x / 3 + C$ .
Explain whether x exp(x squared) and x exp(x) should use the same first method.
2 marks
The derivative of x squared supplies the first factor x.
Differentiating x simplifies the second product without such a match.
No. Substitution is efficient for the first; parts is efficient for the second.
Verify that $e^{x} (x^{2} - 2 x + 2)$ integrates $x^{2} e^{x}$ .
2 marks
Differentiate by the product rule.
The bracket sum simplifies to x squared.
It does.

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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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