Selecting/combining methods | Year 12

01 / LEARN

Understand the idea.

Ask what differentiation would create

Inspect the structure before choosing a method. An inner function accompanied by its derivative suggests substitution. A product with a factor that simplifies under differentiation suggests parts. For a rational expression, first check degree, factorisation, and possible partial fractions.

Useful algebra may reveal a familiar pattern. Completing a square transforms $x^{2} + 6 x + 13$ into ${(x + 3)}^{2} + 4$ , making an arctangent rule visible. The goal is a simpler integral, rather than using a technique because of the expression's appearance.

Explain why a method fits

Explain which feature makes the method work. For $x e^{x^{2}}$ , the factor $x$ is proportional to the derivative of $x^{2}$ , so substitution removes both parts together. Integration by parts is more useful when no such chain-rule match is present.

Different methods can produce antiderivatives that look different but differ only by a constant on an interval. Compare their derivatives to verify equivalence. For a definite integral, also check continuity, sign, and a rough bound before trusting a numerical result.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

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

Worked solution

1
Identify the reverse chain rule.
$u = x^{2}, d u = 2 x d x, x d x = \frac{1}{2} d u$
The polynomial factor is proportional to the derivative of the exponent, making substitution efficient.
2
Integrate after substitution.
$\int x e^{x^{2}} d x = \frac{1}{2} \int e^{u} d u = \frac{1}{2} e^{u} + C$
The composite exponential becomes a basic exponential in u.
3
Substitute back and check.
$\begin{array}{c} \frac{1}{2} e^{x^{2}} + C \\ \frac{d}{d x} (\frac{1}{2} e^{x^{2}}) = \frac{1}{2} e^{x^{2}} (2 x) = x e^{x^{2}} \end{array}$
The derivative confirms that the outside one half cancels the chain factor.

Answer

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

EXAMPLE 02

Worked example 2

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

Worked solution

1
Complete the square.
$x^{2} + 6 x + 13 = {(x + 3)}^{2} + 4$
This writes the denominator as a sum of two squares.
2
Shift the variable.
$u = x + 3, d u = d x, \int \frac{d x}{x^{2} + 6 x + 13} = \int \frac{d u}{u^{2} + 2^{2}}$
The shift isolates the standard arctangent pattern without an extra differential factor.
3
Apply the inverse-trigonometric rule.
$\frac{1}{2} arctan (\frac{u}{2}) + C = \frac{1}{2} arctan (\frac{x + 3}{2}) + C$
The standard formula supplies an outside factor of one over 2.
4
Check the derivative.
$\frac{d}{d x} [\frac{1}{2} arctan (\frac{x + 3}{2})] = \frac{1}{{(x + 3)}^{2} + 4}$
The recovered denominator is the original positive quadratic, so the result holds for every real x.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

Which is shorter for integral ln x / x?

Show answer & reasoning

Substitution u=ln x.

For $u = ln x$ , $d u = \frac{1}{x} d x$ , so $\int \frac{ln x}{x} d x$ becomes $\int u d u$ on $x > 0$ . The derivative factor is already present.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Integrate $\frac{ln x}{x}$ for $x > 0$ .
2 marks
Integrate $\frac{x}{\sqrt{1 - x^{2}}}$ on $- 1 < x < 1$ .
2 marks
Integrate $x e^{2 x}$ .
2 marks
Integrate ${cos}^{2} x sin x$ .
2 marks
Explain whether $x e^{x^{2}}$ and $x e^{x}$ 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 $\frac{ln x}{x}$ for $x > 0$ .
2 marks
Set $u = ln x$ .
Integrate $u d u$ .
$\frac{{(ln x)}^{2}}{2} + C$ .
Integrate $\frac{x}{\sqrt{1 - x^{2}}}$ on $- 1 < x < 1$ .
2 marks
Use $u = 1 - x^{2}$ .
Then $x d x = - \frac{d u}{2}$ .
$- \sqrt{1 - x^{2}} + C$ .
Integrate $x e^{2 x}$ .
2 marks
Use parts with $u = x$ and $v = \frac{e^{2 x}}{2}$ .
Subtract the remaining exponential integral.
$e^{2 x} (\frac{x}{2} - \frac{1}{4}) + C$ .
Integrate ${cos}^{2} x sin x$ .
2 marks
Use $u = cos x$ .
The negative differential changes the sign.
$- \frac{{cos}^{3} x}{3} + C$ .
Explain whether $x e^{x^{2}}$ and $x e^{x}$ should use the same first method.
2 marks
The derivative of $x^{2}$ 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 $x^{2}$ .
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.

Official curriculum source ↗

In this lesson

Understand the idea.

Ask what differentiation would create

Explain why a method fits

See the method in action.

Worked example 1

Identify the reverse chain rule.

Integrate after substitution.

Substitute back and check.

Worked example 2

Complete the square.

Shift the variable.

Apply the inverse-trigonometric rule.

Check the derivative.

Try it yourself.

Check your understanding

Keep practising