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Quick integration map
Core Formulae
Substitution Examples
Integration by Parts
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.
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.
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).
Core Formulae
Reverse of differentiation rules (power, exponential, trigonometric, logarithmic).
Integration by substitution: choose u=g(x) to simplify integrand, rewrite dx=g′(x)du.
Integration by parts: ∫udv=uv−∫vdu.
Partial fractions: decompose rational functions into simpler fractions before integrating.
Substitution Examples
Example -- Trigonometric substitution
Evaluate ∫x2+12xdx.
Let u=x2+1⟹du=2xdx.
Integral becomes ∫u1du=ln∣u∣+C=ln∣x2+1∣+C.
Example -- Exponential substitution
Evaluate ∫xex2dx.
Let u=x2⟹du=2xdx.
Integral =21∫eudu=21ex2+C.
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.
Order
Pick this first when
I
an Inverse trigonometric term (e.g. arctanx) appears
L
a Logarithmic term (e.g. lnx) is present
A
an Algebraic factor (polynomial or rational) remains
T
Trigonometric or hyperbolic functions turn up
E
finally, the Exponential factor eg(x)
Example -- xcosx
Take u=xdv=cosxdx.
∫xcosxdx=xsinx−∫sinxdx=xsinx+cosx+C.
Example -- xex
Take u=xdv=exdx.
∫xexdx=xex−∫exdx=xex−ex+C=ex(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.
Use sin2x=21(1−cos2x), cos2x=21(1+cos2x), and sinxcosx=21sin2x.
For integrals of secx or tanx, multiply numerator and denominator to match derivatives.
Example -- ∫sec2x
Recognise derivative of tanx∫sec2xdx=tanx+C.
Improper Integrals and Convergence
Evaluate limits for infinite bounds or singularities:
∫1∞xp1dx converges iff p>1.
For improper points inside interval, split integral and take limits.
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∣x2+1∣+C, not just ln∣x2+1∣.
Incorrect differential in substitution: After letting u=g(x), students often substitute du without dividing by g′(x). Always write dx=g′(x)du
Wrong assignment of u and dv in integration by parts: Choosing u=ex and dv=lnxdx
Missing the ln∣f(x)∣ pattern: When the numerator is a scalar multiple of f′(x), the integral is kln∣f(x)∣+C
Sign errors with trigonometric integrals: ∫sinxdx=−cosx+C and ∫sec2xdx=tanx+C
Frequently asked questions
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. (x−2)(x+1)3x+5 - 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 ∫tanxdx=ln∣secx∣+C and ∫sec2xdx=tanx+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.
