---
title: "H2 Math Calculus: Differentiation, Integration & Maclaurin"
description: "Calculus is the largest Pure strand in H2 Math, and it builds on — rather than repeats — O-Level calculus. This guide covers the H2-specific techniques: Maclaurin, integration methods, and differential equations."
author: "Gabriel"
author_url: "https://ancourage.academy/authors/gabriel"
published_at: 2026-06-11
modified_at: 2026-06-11
category: "teaching"
tags: ["Mathematics", "JC", "A-Level", "H2 Mathematics", "Calculus", "Singapore", "Exam Tips"]
canonical: "https://ancourage.academy/articles/h2-math-calculus-differentiation-integration-guide-singapore"
source: "https://ancourage.academy/articles/h2-math-calculus-differentiation-integration-guide-singapore"
language: "en-SG"
word_count: 1724
reading_time: "PT9M"
cover_image: "https://ancourage.academy/academic-pic/IMG_8811.jpg"
reviewed_by: "Min Hui"
---

# H2 Math Calculus: Differentiation, Integration & Maclaurin

Calculus is the largest Pure strand in H2 Math, and it builds on — rather than repeats — O-Level calculus. This guide covers the H2-specific techniques: Maclaurin, integration methods, and differential equations.

**Calculus is the largest strand of H2 Mathematics Pure content, and the key to mastering it is recognising that the H2 syllabus assumes your O-Level calculus and then builds new techniques on top — Maclaurin series, advanced integration, and differential equations — rather than re-teaching the basics.** Students who try to relearn differentiation from scratch waste time; the marks are in the new methods. This guide is from [Ancourage Academy](https://ancourage.academy/academy), whose [JC Mathematics tuition](https://ancourage.academy/courses/academy/jc/mathematics) teaches H2 calculus technique by technique in small groups of 3–6.

This is a single-topic deep-dive into the calculus block our [H2 Mathematics JC guide](https://ancourage.academy/articles/h2-mathematics-jc-guide-singapore) only names, completing the Pure Maths series with our [vectors](https://ancourage.academy/articles/h2-math-vectors-lines-planes-guide-singapore), [statistics](https://ancourage.academy/articles/h2-math-statistics-permutations-probability-guide-singapore), and [complex numbers](https://ancourage.academy/articles/h2-math-complex-numbers-guide-singapore) guides. The O-Level foundations it assumes are covered in our [A-Maths calculus guide](https://ancourage.academy/articles/a-maths-calculus-differentiation-integration-singapore).

**If H2 calculus is where your child's marks slip, Ancourage Academy's [JC2 H2 Mathematics programme](https://ancourage.academy/courses/academy/jc/jc2/h2-maths) builds each technique systematically — [book a free trial class (usually $18)](https://ancourage.academy/trial-class) for a diagnostic assessment.**

## Where Does Calculus Sit in H2 Math (9758)?

**In H2 Mathematics (9758), calculus is Topic 5 of the Pure Mathematics section, examinable in Paper 1 and in Paper 2 Section A — and the O-Level basics (differentiating x raised to a power, sin, cos, e^x and ln x; the chain, product and quotient rules; stationary points; basic integration) are listed as assumed knowledge, not re-taught.** The [SEAB 9758 syllabus](https://www.seab.gov.sg/gce-a-level/a-level-syllabuses-examined-for-school-candidates-2026/) organises Topic 5 into five parts: differentiation, Maclaurin series, integration techniques, definite integrals, and differential equations. A graphing calculator is assumed throughout, and the List of Formulae (MF27) provides the standard integration and series results.

## How Does H2 Differentiation Go Beyond O-Level?

**H2 differentiation extends the O-Level rules to functions defined implicitly or parametrically, and uses them for tangents, normals, and the nature of stationary points.**

-   **Implicit differentiation:** differentiate each term with respect to x, treating y as a function of x, then collect dy/dx. Used when y cannot be made the subject.
-   **Parametric differentiation:** when x and y are both given in terms of a parameter t, dy/dx = (dy/dt) ÷ (dx/dt).
-   **Tangents and normals:** including for implicit and parametric curves — find the gradient, then the line equation.
-   **Stationary points and connected rates of change:** classify points using the first or second derivative test, and link rates with dy/dt = (dy/dx)(dx/dt).

Two scoping points worth noting: the second derivative of a parametrically-defined function is excluded, and non-stationary points of inflexion are excluded — only stationary points (maxima, minima, and stationary points of inflexion) are classified for their nature.

## What Is a Maclaurin Series in H2 Math?

**A Maclaurin series expresses a function as an infinite polynomial, and H2 Math requires you to recognise five standard series, derive the first few terms of others, and use the small-angle approximations.**

| Function | Standard Maclaurin series (first terms) |
| --- | --- |
| (1 + x)ⁿ | 1 + nx + \[n(n−1)/2!\]x² + … (for rational n) |
| eˣ | 1 + x + x²/2! + x³/3! + … |
| sin x | x − x³/3! + x⁵/5! − … |
| cos x | 1 − x²/2! + x⁴/4! − … |
| ln(1 + x) | x − x²/2 + x³/3 − … |

You must be able to derive the first few terms of a new series three ways — by repeated differentiation (for example, sec x), by repeated implicit differentiation, and by combining the standard series (for example, eˣcos 2x). The syllabus also requires the range of values of x for which a standard series converges (its range of validity), and the small-angle approximations: sin x ≈ x, cos x ≈ 1 − ½x², and tan x ≈ x. Deriving the general term of a series is excluded — only the first few terms are required.

## What Integration Techniques Does H2 Math Add?

**H2 integration adds two general methods — integration by parts and integration by a given substitution — plus a set of standard forms that appear on the formula list.**

-   **By parts:** for products such as x·eˣ or x·sin x, using ∫u dv = uv − ∫v du. The "LIATE" ordering helps choose u.
-   **By a given substitution:** the substitution is provided in the question — your task is to carry it through, including changing the limits for definite integrals.
-   **Recognising f′(x)·\[f(x)\]ⁿ and f′(x)·e^f(x):** these reverse the chain rule directly, including the n = −1 case that gives a logarithm.
-   **Standard forms (given in MF27):** for example, ∫1/(a² + x²) dx = (1/a)tan⁻¹(x/a) and ∫1/√(a² − x²) dx = sin⁻¹(x/a). The skill is recognising which form a question matches.

Reduction formulae are excluded, and substitution is always by a given substitution rather than one you must invent.

## How Do You Find Areas and Volumes of Revolution in H2 Math?

**The definite integral gives areas and volumes — and a frequent under-practised point is that H2 Math examines the volume of revolution about both the x-axis and the y-axis.**

-   **Area under a curve, between a curve and a line, or between two curves;** including regions below the x-axis (where the integral is negative).
-   **Volume of revolution about the x-axis:** V = π∫y² dx.
-   **Volume of revolution about the y-axis:** V = π∫x² dy — the case students most often forget exists.

Areas and volumes of revolution for parametrically-defined curves are excluded, as are arc length and the surface area of a solid of revolution. Definite integrals can also be approximated on the graphing calculator.

## What Differential Equations Are in H2 Math?

**H2 differential equations are restricted to the separable type, dy/dx = f(x)g(y), solved by separating variables and integrating both sides — and the harder marks are in formulating and interpreting them from real situations.**

The method: rearrange to separate the x and y terms, integrate both sides, include the arbitrary constant, then use a boundary or initial condition to find the particular solution. A question may give a substitution that reduces a non-separable-looking equation to the separable form. Modelling questions — population growth, radioactive decay, Newton's law of cooling — ask you to formulate the equation from the situation and interpret the solution in context. Note the boundaries: the integrating-factor method, and second-order or non-separable equations, belong to other syllabuses, not 9758 — though the simplest case dy/dx = f(x) is just the separable form with g(y) = 1.

## What Are the Most Common H2 Calculus Mistakes?

**Most H2 calculus mistakes come from choosing the wrong method, importing excluded techniques, or forgetting syllabus-specific applications.**

| Mistake | Why It Happens | How to Fix It |
| --- | --- | --- |
| Forgetting y-axis volumes | Assuming revolution is only about the x-axis | Check the axis named; use V = π∫x² dy for the y-axis |
| Not changing limits in substitution | Substituting the variable but keeping x-limits | Convert the limits to the new variable before evaluating |
| Wrong constant in differential equations | Skipping the boundary condition | Always apply the given condition to find the particular solution |
| Importing out-of-syllabus methods | Using integrating factors or reduction formulae | 9758 uses separable DEs and by-parts/given-substitution only |
| Misreading the standard integration form | Confusing 1/(a²+x²) with 1/√(a²−x²) | Match the denominator's exact shape to the MF27 result |
| Choosing the wrong u in by-parts | Picking the part that gets harder to integrate | Use LIATE — pick u as the earlier type (logs, algebra) so it simplifies |

## How Do You Study H2 Calculus Effectively?

**H2 calculus rewards securing the O-Level foundation first, then layering the new techniques one at a time and drilling recognition — knowing which method a question wants is half the work.**

1.  **Confirm the assumed knowledge:** if O-Level differentiation and basic integration are shaky, fix them first — H2 builds directly on them.
2.  **One technique at a time:** Maclaurin, then by-parts, then substitution, then standard forms, then volumes, then differential equations — mastering each before mixing.
3.  **Drill method recognition:** practise spotting which standard form or method a question maps to, since exams test selection as much as execution.
4.  **Master the graphing calculator workflow:** locating maxima/minima, approximating derivatives and definite integrals are in-syllabus skills.

At Ancourage Academy, our [JC Mathematics programme](https://ancourage.academy/courses/academy/jc/mathematics) teaches H2 calculus technique by technique in small groups of 3–6 at [Bishan](https://ancourage.academy/find-us/bishan) and [Woodlands](https://ancourage.academy/find-us/woodlands). Students who did not take A-Maths should read whether they [need A-Maths for H2 Math](https://ancourage.academy/articles/do-you-need-a-maths-for-h2-math-jc-singapore). Book a [free trial class (usually $18)](https://ancourage.academy/trial-class) for a diagnostic, or [WhatsApp us](https://api.whatsapp.com/send/?phone=6588498106&type=phone_number&app_absent=0) with any questions.

## Common Questions About H2 Math Calculus

### What does H2 Math calculus cover that O-Level A-Maths does not?

H2 calculus assumes the O-Level basics (basic differentiation and integration, the chain, product and quotient rules, stationary points) and adds: implicit and parametric differentiation, Maclaurin series and small-angle approximations, integration by parts and by a given substitution, standard integration forms, areas and volumes of revolution about both axes, and separable differential equations. The O-Level material is assumed knowledge, so revision time should go on these new techniques rather than re-learning the basics.

### How many standard Maclaurin series do I need to know?

Five: the expansions of (1 + x)ⁿ for rational n, eˣ, sin x, cos x, and ln(1 + x). They are provided in the MF27 List of Formulae, but you must apply them fluently — for example, combining them to find the series of a product like eˣcos 2x — and know each one's range of validity. You also need the three small-angle approximations: sin x ≈ x, cos x ≈ 1 − ½x², and tan x ≈ x. Deriving the general term of a series is not required.

### What type of differential equations are in H2 Math?

Only separable differential equations of the form dy/dx = f(x)g(y), solved by separating the variables and integrating both sides, then applying a boundary condition for the particular solution. A question may provide a substitution that reduces an equation to this form. Modelling questions — population growth, radioactive decay, Newton's law of cooling — require you to formulate and interpret the equation. Integrating factors, second-order equations, and other non-separable methods are not in the 9758 syllabus.

### Is volume of revolution about the y-axis tested?

Yes. H2 Math examines the volume of revolution about both the x-axis (V = π∫y² dx) and the y-axis (V = π∫x² dy). The y-axis case is the one students most often overlook, because it is less frequently drilled. Volumes for parametrically-defined curves are excluded, so all area and volume-of-revolution questions use Cartesian curves.

Related: [H2 Mathematics JC Guide](https://ancourage.academy/articles/h2-mathematics-jc-guide-singapore) · [H2 Vectors Guide](https://ancourage.academy/articles/h2-math-vectors-lines-planes-guide-singapore) · [H2 Statistics & Probability Guide](https://ancourage.academy/articles/h2-math-statistics-permutations-probability-guide-singapore) · [H2 Complex Numbers Guide](https://ancourage.academy/articles/h2-math-complex-numbers-guide-singapore) · [A-Maths Calculus Guide](https://ancourage.academy/articles/a-maths-calculus-differentiation-integration-singapore)

## Sources

- [Mathematics (Syllabus 9758) — 2026 Examination (seab.gov.sg)](https://www.seab.gov.sg/gce-a-level/a-level-syllabuses-examined-for-school-candidates-2026/) — Singapore Examinations and Assessment Board
- [A-Level Curriculum and Subject Syllabuses (moe.gov.sg)](https://www.moe.gov.sg/post-secondary/a-level-curriculum-and-subject-syllabuses) — Ministry of Education, Singapore
- [List of Formulae and Statistical Tables (MF27) for A-Level Mathematics (seab.gov.sg)](https://www.seab.gov.sg/gce-a-level/) — Singapore Examinations and Assessment Board