H2 MATHEMATICS · INTEGRATION

Read the structure.
Then choose the technique.

A technique-selection guide for Singapore A-Level H2 Mathematics syllabus 9758. Recognise reverse chain rule, integration by parts and helpful algebraic preparation.

Study the worked example Try an integration question

Technique selection
Exact antiderivatives
Fast differentiation checks

01Differentiate to check A correct antiderivative returns the original integrand.

Built around 9758 H2 MathematicsSubstitution, integration by parts and definite integrals

Technique before algebraIdentify the structural clue before expanding or simplifying

THE SELECTION HABIT

Ask what differentiation would produce

Integration is reverse differentiation. A function beside its derivative suggests substitution; a product whose factors simplify differently often suggests integration by parts.

Before committing, consider whether division, completing the square or a trigonometric identity will reveal a standard form.

THE TECHNIQUE SELECTOR

Match the visible structure

Composite function?

Look for an inner function and a multiple of its derivative.

\int f' (x) g (f (x)) d x

Product?

Choose u to become simpler when differentiated.

\int u d v = u v - \int v d u

Rational expression?

Divide or decompose before integrating term by term.

\frac{x^{2} + 1}{x + 1}

Definite integral?

Transform the limits with a substitution or return carefully to x.

\int_{a}^{b} f (x) d x

WORKED EXAMPLE

A product points to integration by parts.

Find $\int x e^{x} d x$ .

EXAM HABIT

Choose u as the factor that simplifies when differentiated.

1
Choose u and dv
$u = x, d v = e^{x} d x$
2
Differentiate and integrate
$d u = d x, v = e^{x}$
3
Apply the formula
$\int x e^{x} d x = x e^{x} - \int e^{x} d x$
4
Simplify and check
$= e^{x} (x - 1) + C$
Differentiating the result gives x eˣ.

Answereˣ(x − 1) + C

COMMON MISTAKES

What to catch before the examiner does

Forcing substitution

An inner expression is not enough; its derivative must also be available or obtainable.

Poor choice of u

In integration by parts, choose the factor that becomes simpler under differentiation.

Dropping + C

An indefinite integral represents a family of antiderivatives.

Not checking

Differentiate the final expression to expose sign and coefficient errors quickly.

PRACTISE WITH FEEDBACK

Explain why the technique fits.

Submit your selection and working to Integrand for step-specific feedback.

Try Integrand

1
Inspect
Identify the structure.
2
Transform
Apply the chosen method.
3
Check
Differentiate the result.

QUICK QUESTIONS

H2 integration FAQ

When should I use integration by parts?

It is especially useful for products where differentiating one factor simplifies it, such as x eˣ, x sin x or x ln x.

How do I check an indefinite integral?

Differentiate your answer. It should simplify to the original integrand.

Is this guide within the H2 syllabus?

Yes. Integration techniques including substitution and integration by parts are part of Singapore A-Level H2 Mathematics syllabus 9758.

READY TO PRACTISE?

Choose a technique. Get feedback on the working.

Open Integrand

Read the structure.Then choose the technique.

Ask what differentiation would produce

Match the visible structure

Composite function?

Product?

Rational expression?

Definite integral?

A product points to integration by parts.

Choose u and dv

Differentiate and integrate

Apply the formula

Simplify and check

What to catch before the examiner does

Forcing substitution

Poor choice of u

Dropping + C

Not checking

Explain why the technique fits.

Inspect

Transform

Check

H2 integration FAQ

Explore another A-Level Maths guide

Vectors

Differential equations

Inequalities

Choose a technique. Get feedback on the working.

Read the structure.
Then choose the technique.