Singapore Mathematics | International Edition: Integration

Singapore Math guide to integration, the upper secondary Additional Mathematics topic that reverses differentiation and measures area, for families and teachers outside Singapore.

Differentiation takes a function and finds its gradient.

Integration starts from the gradient and rebuilds the function.

In Singapore Mathematics, students in G3 Additional Mathematics (Secondary 3 to 4) learn integration as the reverse of differentiation. They integrate a listed set of functions, evaluate definite integrals and use them to find areas, including areas below the x-axis.

The same skills are then used in kinematics, and they are developed further at A-Level.

This guide is useful for families and teachers who want to:

• understand what the Additional Mathematics syllabus includes and excludes;
• teach integration as undoing a derivative, then as area;
• prepare a student who is moving into a Singapore secondary school; or
• support a student using Singapore Math materials in another country.

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What Integration Means in Singapore Mathematics

If dy/dx = 2x, then y could be x², or x² + 5, or x² − 1. Every one of these has the derivative 2x.

So the indefinite integral of 2x is x² + c, where c is an arbitrary constant. A known point on the curve fixes the value of c.

A definite integral has limits, such as from x = 1 to x = 3. It gives a number rather than a function.

That number is linked to area. For a curve above the x-axis, the definite integral equals the area between the curve, the x-axis and the two vertical lines at the limits.

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Where Integration Sits in the Singapore Syllabus

Integration belongs to the Calculus strand of G3 Additional Mathematics, after differentiation.

Students learn:

• integration as the reverse of differentiation;
• integration of xⁿ for any rational n, sin x, cos x, sec²x and e^x, with constant multiples, sums and differences;
• integration of (ax + b)ⁿ for any rational n, sin (ax + b), cos (ax + b) and e^{ax + b};
• the definite integral as the area under a curve, and evaluating definite integrals;
• the area of a region bounded by a curve and line(s); and
• areas of regions below the x-axis.

The area of a region between two curves is excluded.

Course	What students do
G3 Additional Mathematics, Sec 3 to 4	Listed integrals, definite integrals, areas bounded by a curve and lines, areas below the x-axis
A-Level H2 Mathematics	Further techniques such as integration by substitution and by parts; areas between two curves; volumes of revolution

The integral results are not on the Additional Mathematics formula list, so students must know them.

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How Singapore Teaches Integration

1. Reverse a derivative

Students differentiate x³ to get 3x², then ask what function gives 3x². They see why every answer needs + c.

2. Build the power rule

∫xⁿ dx = x^{n + 1}/(n + 1) + c, for n ≠ −1. Roots and reciprocals are rewritten first, so ∫√x dx = ∫x^1/2 dx = (2/3)x^3/2 + c.

3. Add the listed functions

∫cos x dx = sin x + c, ∫sin x dx = −cos x + c, ∫sec²x dx = tan x + c and ∫e^x dx = e^x + c. Each comes from a derivative students already know.

4. Handle ax + b

For (ax + b)ⁿ, sin (ax + b), cos (ax + b) and e^{ax + b}, students integrate as usual and divide by a. Differentiating the answer checks it.

5. Evaluate definite integrals

Students substitute the upper limit, then the lower limit, and subtract. The constant c cancels.

6. Find areas

For a region below the x-axis, the integral is negative, so its size is taken as the area. For a region bounded by a curve and a straight line, students combine an integral with the area of a triangle or trapezium.

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Worked Examples

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Common Mistakes

• Forgetting + c. An indefinite integral without + c is incomplete. Fix: write + c every time, then use any given point to find it.
• Not dividing by a. ∫(2x + 1)⁴ dx is (2x + 1)⁵/10 + c, not (2x + 1)⁵/5 + c. Fix: differentiate the answer to check.
• Mixing up signs for sine and cosine. ∫sin x dx = −cos x + c. Fix: recall that the derivative of cos x is −sin x and reverse it.
• Integrating across the x-axis in one go. Regions above and below the axis cancel. Fix: sketch the curve, find where it crosses the axis and integrate each part separately.
• Integrating a product term by term. ∫x(x + 1) dx is not (x²/2)(x²/2 + x). Fix: expand first, then integrate each term.
• Using degrees in the limits. For trigonometric integrals, limits such as π/2 must be in radians. Fix: set the calculator to radians.

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Teaching Integration Outside Singapore

• Notation. Singapore materials use the same ∫ f(x) dx notation found in most English-language books, with + c for the arbitrary constant.
• What is excluded. The area between two curves is not in Additional Mathematics. Areas bounded by one curve and straight lines are.
• Timing. Integration here sits in the upper secondary elective. The Singapore Math Grade Levels guide explains why grade-by-grade comparisons are unreliable.
• Prerequisites. Students need secure differentiation, indices, radians and exact trigonometric values.
• Calculators. An approved calculator may be used in both papers. Students should still show the integral, the limits and the subtraction.

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How to Practise Integration at Home

1. Differentiate three functions, then reverse each one as an integral.
2. Rewrite three roots or fractions as powers and integrate them.
3. Integrate one expression of each type: (ax + b)ⁿ, sin (ax + b), cos (ax + b) and e^{ax + b}.
4. Check every indefinite answer by differentiating it.
5. Sketch one curve, mark the region and find its area, splitting at the x-axis if needed.
6. Once a week, find a curve’s equation from its gradient function and a point.

Sketching first prevents most area errors.

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Try These

1. Find ∫(3x² + 2) dx.
2. Find ∫(2x + 1)⁴ dx.
3. Evaluate the definite integral of cos x from x = 0 to x = π/2.
4. Find the area of the region bounded by y = 4 − x² and the x-axis.

Answers

1. x³ + 2x + c.
2. (2x + 1)⁵/10 + c. Raise the power to 5, divide by 5 and divide by 2.
3. 1, because sin (π/2) − sin 0 = 1 − 0 = 1.
4. 32/3 square units. The curve meets the x-axis at x = −2 and x = 2, and [4x − x³/3] from −2 to 2 gives (8 − 8/3) − (−8 + 8/3) = 32/3.

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What Comes Next

Integration completes the Calculus strand of Additional Mathematics.

• Kinematics integrates acceleration to get velocity, and velocity to get displacement.
• Differentiation is the check for every integral, so keep it fresh.
• At A-Level, H2 Mathematics adds techniques such as substitution and integration by parts, and areas between two curves.

A student who checks each integral by differentiating it builds accuracy that lasts.

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Frequently Asked Questions

When do Singapore students learn integration?

In G3 Additional Mathematics, the upper secondary elective taken in Secondary 3 to 4, after differentiation.

Which functions do students integrate?

xⁿ for any rational n, sin x, cos x, sec²x and e^x, and the forms (ax + b)ⁿ, sin (ax + b), cos (ax + b) and e^{ax + b}, with multiples, sums and differences.

Is the area between two curves included?

No. It is excluded from Additional Mathematics. It appears later in H2 Mathematics.

Why does an area below the x-axis come out negative?

Below the axis, y is negative, so the integral is negative. The area is the size of that value.

Are integration by parts and substitution in the syllabus?

They are not listed in Additional Mathematics. Both appear in H2 Mathematics.

Is integration on the formula list?

No. The integral results are not on the formula list, so students learn them.

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Helpful Reading

• Singapore Mathematics | International Edition: Differentiation
• Singapore Mathematics | International Edition: Kinematics
• Singapore Mathematics | International Edition: Trigonometric Identities and Equations
• Why Does Integration Feel Like Differentiation Backwards?
• Secondary 3 Additional Mathematics | Integration
• Secondary 3 Additional Mathematics | Definite Integrals and Area
• How to Make Sense of the SEC G3 Additional Mathematics Syllabus
• SEAB G3 Additional Mathematics Syllabus K341

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Integration in Singapore Mathematics

Reverse the derivative.

Always add c.

Divide by a.

Sketch before finding an area.

Split at the x-axis.

Taught this way, integration stops feeling like differentiation backwards.

It becomes the mathematics of building up a whole from its rate of change.

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Ask Bukit Timah Tutor

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If your child is learning Singapore Mathematics at home, in an international school or in a Singapore school, and you are unsure where to begin with this topic, send us a short message with the student’s level and one example of the work.

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Singapore Mathematics | International Edition is a Bukit Timah Tutor guide series for families and teachers outside Singapore. It is not a textbook and is not affiliated with MOE, SEAB or any publisher. Syllabus details were checked against MOE and SEAB documents in October 2026. Always confirm current requirements with your school.

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