H2 Maths Integration Formula Sheet | Standard Forms & Techniques

Q: What does H2 Maths Notes (JC 1-2): 5.3) Integration Techniques cover?
A: Substitution, parts, partial fractions, and special integrals for H2 Maths Topic 5.3.

Download: Get the H2 Maths Integration formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

Before you revise
Build a flowchart for integration: recognise the pattern (substitution, parts, partial fractions, trigonometric identities) before diving in. Always write differential substitutions explicitly.

• Reverse differentiation: Basic forms may already fit.
• Substitution, parts, or partial fractions: The integrand's structure chooses the method.
• Check by differentiating: It catches missing constants, signs, and factors.

Concrete example: For $\int \frac{2x}{x^2+1},dx$, let $u=x^2+1$

Before choosing a method, ask what the integrand is trying to hide. Do this screening step before expanding, splitting, or substituting.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 5.3 scope is within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

Formulas at a glance

Every result the 9758 syllabus expects you to recall or apply, on one screen. Standard integrals listed in MF27 are marked (MF27); everything else must be memorised or derived. Worked examples for each technique appear in the sections below.

Standard integrals

Integration by substitution

Integration by parts

Partial fractions

Each linear-factor term integrates to a $\ln\lvert \cdot \rvert$ form.

Trigonometric identities for integration

Improper integrals

Core Formulae

• Reverse of differentiation rules (power, exponential, trigonometric, logarithmic).
• Integration by substitution: choose $u = g(x)$ to simplify integrand, rewrite $\mathrm{d}x = \dfrac{\mathrm{d}u}{g'(x)}$

Method Switch Checkpoint

Run this quick sequence before committing to a long technique.

Common trap: do not use integration by parts just because the integrand is a product. If one factor is already the derivative of an inside function, substitution is usually shorter.

Substitution Examples

Example -- Trigonometric substitution

Evaluate $\int \dfrac{2x}{x^2 + 1} \space \mathrm{d}x$.

1. Let $u = x^2 + 1 \implies \mathrm{d}u = 2x \space \mathrm{d}x$.
2. Integral becomes $\int \dfrac{1}{u} \space \mathrm{d}u = \ln\lvert u \rvert + C = \ln\lvert x^2 + 1 \rvert + C$

Example -- Exponential substitution

Evaluate $\int x e^{x^2} \space \mathrm{d}x$.

1. Let $u = x^2 \implies \mathrm{d}u = 2x \space \mathrm{d}x$.
2. Integral $= \tfrac{1}{2} \int e^{u} \space \mathrm{d}u = \tfrac{1}{2} e^{x^2} + C$

Definite-substitution checkpoint

When substitution appears inside a definite integral, choose one consistent route before applying the bounds.

Worked check: for $\int_0^1 2x(x^2+1)^3,\mathrm{d}x$, let $u=x^2+1$

Misconception check: changing variables changes the meaning of the bounds. A bound of $1$ in $x$ is not automatically a bound of $1$ in $u$.

Integration by Parts

• Choose $u$ to simplify upon differentiation; set $dv$ as remaining factor.
• For repeated parts, loop until integral becomes elementary.

A handy heuristic for choosing $u$ is the ILATE/LIATE mnemonic-use it to decide which factor to differentiate so the algebra keeps simplifying. It is widely taught even though the official H2 syllabus only specifies the integration-by-parts technique itself, so treat ILATE/LIATE as planning advice rather than examinable content.

Repeated-parts checkpoint

Before starting integration by parts, decide what should happen after one round. This prevents a long chain of parts steps with no target.

Worked check: let $I=\int e^x\cos x,\mathrm{d}x$. One round gives

$$I=e^x\cos x+\int e^x\sin x,\mathrm{d}x.$$

A second round on the remaining integral gives $\int e^x\sin x,\mathrm{d}x=e^x\sin x-I$, so

$$2I=e^x(\sin x+\cos x),\quad I=\frac{e^x(\sin x+\cos x)}{2}+C.$$

Misconception check: when the original integral returns, do not keep integrating by parts forever. Move it to the left-hand side and divide.

Example -- $x \cos x$

Take $u = x \space dv = \cos x \space \mathrm{d}x$. $$\int x \cos x \space \mathrm{d}x = x \sin x - \int \sin x \space \mathrm{d}x = x \sin x + \cos x + C.$$

Example -- $x e^x$

Take $u = x \space dv = e^x \space \mathrm{d}x$. $$\int x e^x \space \mathrm{d}x = x e^x - \int e^x \space \mathrm{d}x = x e^x - e^x + C = e^x(x - 1) + C.$$

Note: reduction formulae are excluded in the 2026 H2 syllabus-use integration by parts directly for each required integral.

Partial Fractions

• Proper rational functions: express as sum of linear/quadratic factors.
• For repeated factors, include each power up to multiplicity.
• For irreducible quadratics, use $\dfrac{Ax + B}{\text{quadratic}}$.

Example -- Partial fraction integral

Evaluate $\int \dfrac{3x + 5}{x^2 - x - 2} \space \mathrm{d}x$.

1. Factor denominator: $(x - 2)(x + 1)$.
2. Decompose: $\dfrac{3x + 5}{(x - 2)(x + 1)} = \dfrac{A}{x - 2} + \dfrac{B}{x + 1}$

Trigonometric Identities

• Use $\sin^2 x = \tfrac{1}{2}(1 - \cos 2x)$, $\cos^2 x = \tfrac{1}{2}(1 + \cos 2x)$

Example -- $\int \sec^2 x$

Recognise derivative of $\tan x \:$ $\int \sec^2 x \space \mathrm{d}x = \tan x + C$.

Improper Integrals and Convergence

• Evaluate limits for infinite bounds or singularities: $$\int_1^\infty \dfrac{1}{x^p} \space \mathrm{d}x \text{ converges iff } p > 1.$$

Improper-integral setup checkpoint

Before integrating, locate every place where the integral stops being ordinary. The setup is often worth more than the antiderivative because one missed limit can change convergence to divergence.

Worked check: $\int_0^2 \dfrac{1}{(x-1)^2},\mathrm{d}x$

Misconception check: an antiderivative formula is not enough for an improper integral. The answer depends on the limit process, not just on substituting the two printed bounds.

Calculator Workflow

• Use a graphing calculator (GC) to confirm antiderivatives by differentiating results.
• Numerical integration (∫f(x)dx) checks definite integrals; still show analytical steps.
• Store substitution results before back-substituting to reduce mistakes.

Exam Watch Points

• Explicitly state substitution and new limits for definite integrals.
• Include constant of integration for indefinite integrals.
• Handle absolute values in logarithms when integrating rational functions.
• Justify convergence when dealing with improper integrals.

Want weekly guided practice on Integration Techniques? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

• Omitting the constant of integration: Every indefinite integral requires $+ C$. Markers deduct a mark for a bare antiderivative - write $\ln\lvert x^2 + 1 \rvert + C$, not just $\ln\lvert x^2 + 1 \rvert$

Frequently asked questions

Is there a formula sheet for H2 Maths integration?
Yes - the "Formulas at a glance" section near the top of this page collects the standard integral forms, substitution setup, integration by parts, partial-fraction templates, trigonometric identities, and improper-integral conditions used in this topic. MF27 provides selected standard integrals such as $\int \tan x , \mathrm{d}x$, but it does not list $\int \sec^2 x , \mathrm{d}x$. It also provides partial-fraction decomposition templates, not the complete decomposition or integration method.

Which papers examine Integration Techniques?
Integration Techniques (Topic 5.3) is examined in both Paper 1 (Pure Mathematics, 100 marks, 3 hours) and Paper 2 Section A (Pure Mathematics, 40 marks). [1] Either paper can feature substitution, integration by parts, or partial-fractions integrals, so you must be fluent across all three methods.

Is integration using partial fractions examinable at H2 level?
Yes. Partial fractions (Topic 1.3) and integration are explicitly linked in the 9758 syllabus. [1] A typical question decomposes a rational integrand - e.g. $\dfrac{3x+5}{(x-2)(x+1)}$ - into partial fractions first, then integrates each term as a logarithm. Both the decomposition and the integration steps must be shown for full marks.

Is a formula sheet provided in the H2 Maths exam?
Yes - MF27 (List of Formulae and Results) is provided for both papers. [1] It includes selected standard integrals such as $\int \tan x \space \mathrm{d}x = \ln(\sec x) + C$ on the stated domain, but it does not list $\int \sec^2 x \space \mathrm{d}x = \tan x + C$

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

• Functions & graphs: Functions · Graphs & Transformations · Equations & Inequalities
• Sequences & series: Sequences & Series
• Vectors: Vectors
• Calculus: Differentiation · Maclaurin Series · Integration Techniques (this page) · Definite Integrals · Differential Equations
• Statistics: Probability · Discrete Random Variables · Normal Distribution · Sampling · Hypothesis Testing · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 5 Calculus (integration techniques, improper integrals, partial fractions, and areas/volumes): SEAB 9758 syllabus PDF

Practice Quiz

Reinforce substitution, parts, and partial-fractions strategies-including improper integral convergence checks.

Quick Revision Checklist

• Identify the right technique (substitution, parts, partial fractions) for each integrand.
• Carry out algebraic manipulations cleanly before integrating.
• Evaluate definite integrals with correct limits and substitution steps.
• Discuss convergence for improper integrals and document limit processes.

Next steps: Stay synced via the H2 Maths notes hub and tackle Topic 5.4 - Definite integrals for area/volume modelling.