Is there a formula sheet for H2 Maths integration?

Yes - the "Formulas at a glance" section near the top of this page collects every result you need: standard integral forms, the substitution setup, the integration by parts formula with ILATE ordering, partial-fraction decomposition templates, the three trigonometric identities for reducing powers, and the convergence condition for improper integrals. Note that MF27 provides some standard integrals (e.g. \$\int \tan x \, \mathrm\{d\}x\$, \$\int \sec^2 x \, \mathrm\{d\}x\$), but techniques such as integration by parts applications, substitution setups, and partial-fraction procedures must be understood and carried out without the sheet.

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 Statistical Tables) is provided for both papers. [1] It includes standard integrals such as \$ \int \tan x \space \mathrm\{d\}x = \ln\lvert \sec x \rvert + C \$ and \$ \int \sec^2 x \space \mathrm\{d\}x = \tan x + C \$. You are still expected to derive results not on MF27 (e.g. integration by parts applications), so practise working without the sheet as well. ---

H2 Maths Integration Formula Sheet | Standard Forms & Techniques

Study guide07 Oct 2025, 00:00 Z

H2 Maths integration formula sheet: standard integrals, substitution setup, integration by parts, partial fractions, and trigonometric identities - every key result on one page,...

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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 $\int \frac{2x}{x^2+1},dx$

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Marcus Pang·Managing Director (Maths)

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SEAB H2 Mathematics (9758) Syllabus 2026

Integrand signal	Method to test first	Check before you commit
A clear inside function and its derivative nearby	Substitution	Can you rewrite every part in terms of the new variable?
A product where one factor simplifies when differentiated	Integration by parts	Does your chosen `u` get simpler after differentiating?
A rational function with a factorable denominator	Partial fractions	Is the numerator degree lower than the denominator degree?
Powers of trigonometric functions	Trigonometric identities	Can an identity turn the integrand into basic sine or cosine forms?
Infinite bounds or a denominator that can become zero	Improper-integral limit check	Have you split the integral at every problem point?

Integrand	Result
$x^n$ ( $n \neq -1$ )	$\dfrac{x^{n+1}}{n+1} + C$
$\dfrac{1}{x}$	$\ln\lvert x \rvert + C$
$e^x$	$e^x + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$ (MF27)
$\tan x$	$\ln\lvert \sec x \rvert + C$ (MF27)
$\dfrac{f'(x)}{f(x)}$	$\ln\lvert f(x) \rvert + C$

Step	What to write
Choose substitution	$u = g(x)$
Rewrite differential	$\mathrm{d}x = \dfrac{\mathrm{d}u}{g'(x)}$
For definite integrals (change limits)	Convert each $x$ -limit to a $u$ -limit via $u = g(x)$

Formula	$\displaystyle\int u , \mathrm{d}v = uv - \int v , \mathrm{d}u$
ILATE order for choosing $u$	Inverse trig → Logarithm → Algebraic → Trig → Exponential
Cyclic case (e.g. $e^x \cos x$ )	Apply parts twice; collect $I$ on one side and divide

Denominator factor	Decomposition form
Distinct linear: $(x-a)(x-b)$	$\dfrac{A}{x-a} + \dfrac{B}{x-b}$
Repeated linear: $(x-a)^2$	$\dfrac{A}{x-a} + \dfrac{B}{(x-a)^2}$
Irreducible quadratic: $ax^2+bx+c$	$\dfrac{Ax+B}{ax^2+bx+c}$

H2 Maths Integration Formula Sheet | Standard Forms & Techniques

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Formulas at a glance

Core Formulae

Method Switch Checkpoint

Substitution Examples

Definite-substitution checkpoint

Integration by Parts

Repeated-parts checkpoint

Partial Fractions

Trigonometric Identities

Improper Integrals and Convergence

Improper-integral setup checkpoint

Calculator Workflow

Exam Watch Points

Common exam mistakes

Frequently asked questions

Other H2 Maths formula sheets

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Practice Quiz

Quick Revision Checklist

Related Posts

Identity	Use
$\sin^2 x = \tfrac{1}{2}(1 - \cos 2x)$	Reduce powers of $\sin x$
$\cos^2 x = \tfrac{1}{2}(1 + \cos 2x)$	Reduce powers of $\cos x$
$\sin x \cos x = \tfrac{1}{2} \sin 2x$	Simplify products of $\sin x$ and $\cos x$

Form	Convergence condition
$\displaystyle\int_1^\infty \dfrac{1}{x^p} , \mathrm{d}x$	Converges if and only if $p > 1$
Infinite bound	Replace with parameter $b$ , then take $\lim_{b\to\infty}$
Singularity inside interval	Split the integral at the problem point; take one-sided limits on each piece

Check	If yes	If no
Does the integrand already match a derivative pair?	Integrate directly and check by differentiating.	Look for a rewrite.
Can a factor be rewritten using algebra or an identity?	Simplify first, then recheck basic forms.	Look for a hidden inside function.
Is there an inside function with its derivative nearby?	Use substitution and rewrite all terms in the new variable.	Check for a product or rational form.
Is it a product where one factor becomes simpler when differentiated?	Use integration by parts, choosing that factor as $u$ .	Check whether partial fractions or limits are needed.
Is it a rational function with a factorable denominator?	Make it proper, decompose, then integrate logarithm terms carefully.	Consider trigonometric identities or an improper-integral limit check.

Route	What to do	When it is cleanest	Common trap
Change the limits	Convert each original $x$ -limit into a new $u$ -limit, then integrate fully in $u$ .	The new limits are simple numbers.	Using old $x$ -limits after changing the integrand to $u$ .
Back-substitute first	Find the antiderivative in $u$ , replace $u$ with the original expression in $x$ , then apply the original limits.	The new limits look awkward or the final expression is easy in $x$ .	Applying limits while the expression is still in the wrong variable.
Check the differential	Rewrite every remaining $x$ and $\mathrm{d}x$ term before integrating.	The integrand contains extra factors outside the inside function.	Cancelling only part of the differential and leaving mixed variables.

Order	Pick this first when
I	an Inverse trigonometric term (e.g. $\operatorname{arctan} x$ ) appears
L	a Logarithmic term (e.g. $\ln x$ ) is present
A	an Algebraic factor (polynomial or rational) remains
T	Trigonometric or hyperbolic functions turn up
E	finally, the Exponential factor $e^{g(x)}$

Pattern	Best move	Stop condition	Common trap
Polynomial times exponential or trigonometric factor	Differentiate the polynomial each round.	The polynomial derivative becomes zero.	Integrating the polynomial factor instead, which makes the algebra grow.
Logarithmic or inverse trigonometric factor alone	Set that factor as $u$ and use $1,\mathrm{d}x$ as $\mathrm{d}v$ .	The remaining integral becomes a rational or algebraic form.	Waiting for an obvious second factor before using parts.
Exponential times sine or cosine	Use parts twice, then collect the original integral on one side.	The original integral reappears.	Treating the repeated integral as failure instead of solving for it.

Problem point	First move	What to write	Common trap
Upper or lower bound is infinite	Replace infinity with a parameter.	$\lim_{b\to\infty}\int_a^b f(x),\mathrm{d}x$ , or the matching negative-infinity limit.	Substituting infinity directly into the antiderivative.
Denominator is zero at an endpoint	Move the endpoint slightly inside the interval.	$\lim_{a\to c^+}\int_a^d f(x),\mathrm{d}x$
Denominator is zero inside the interval	Split the integral at that point.	$\int_a^c f(x),\mathrm{d}x+\int_c^b f(x),\mathrm{d}x$
Antiderivative limit does not settle to a finite value	State divergence.	The integral diverges if any required limit is infinite or does not exist.	Averaging one convergent side with one divergent side.