Intro
The HSC Integrals booklet is a free Mathematics Extension 1 and Extension 2 resource with 63 worked problems covering substitution, integration by parts, partial fractions, reduction formulae, volumes of solids, and trigonometric substitutions. It is written for Extension 2 students aiming to master integration and advanced problem-solving and designed for structured HSC revision on Vu's Maths Hub.
This deep-dive introduces HSC Integrals — browser-readable, aligned with the NESA syllabus.
Summary
The HSC Integrals booklet offers a fundamentals review, 63 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Integrals — no account required. Use this post to plan how to work through a 82-page booklet efficiently.
What is this booklet?
This booklet is written for Extension 2 students aiming to master integration and advanced problem-solving.
Focus: substitution, integration by parts, partial fractions, reduction formulae, volumes of solids, and trigonometric substitutions.
Topics covered:
- Substitution
- Integration by parts
- Partial fractions
- Reduction formulae
- Volumes of solids
- Definite integral properties
- Trigonometric substitutions
How to use it:
- Review fundamentals first
- Part 1 without solutions on first pass
- Part 2: try, then upside-down hint
- Practise from memory; use appendices for technique lookup
Approximately 82 pages, CC BY 4.0, readable at HSC Integrals.
Key fundamental reviews
The booklet opens with a dedicated Fundamentals Review — read it before Part 1. It is a compact reference for notation, formulas, and reasoning patterns:
- Substitution and change of limits
- Integration by parts and LIATE heuristics
- Partial fractions for rational integrands
- Reduction formulae and recursive integration
- Volumes of solids of revolution
- Definite integral properties and symmetry
Why fundamentals matter
Substitution and change of limits — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Integration by parts and LIATE heuristics — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Partial fractions for rational integrands — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Reduction formulae and recursive integration — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Volumes of solids of revolution — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.
Students who skip this section often repeat the same algebra errors in Part 2. Treat HSC Integrals fundamentals as a closed-book quiz first.
Problems and how to use them
The HSC Integrals booklet packs 63 practice problems into roughly 82 pages — well beyond a single textbook chapter. Each item includes worked solutions; many include Takeaways that highlight the method to reuse in exams.
Overall structure
Part 1 — detailed solutions: easy, medium, and hard problems with full solutions.
Part 2 — hint-based fluency: matching tiers with upside-down hints.
Use Part 1 to learn how complete NSW HSC working is written. Use Part 2 in the fortnight before trials — hints are upside-down so you attempt first.
Part 1 (15 problems)
Easy (5 problems)
This tier contains 5 problems aimed at foundational fluency. Representative work includes "Partial Fractions" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Partial Fractions
- Integration by Parts
- Reverse Chain Rule
- Algebraic Substitution
- Definite Integral Property
Hard (5 problems)
This tier contains 5 problems aimed at extension and synthesis. Representative work includes "Advanced Reduction Formula with Induction" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Advanced Reduction Formula with Induction
- Reduction Formula with Factorial Series
- Substitution Proof
- Applications - Inclined Plane Dynamics
- Volumes of Revolution with Ratio
Medium (5 problems)
This tier contains 5 problems aimed at exam-standard reasoning. Representative work includes "Reduction Formula for Powers of Cotangent" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Reduction Formula for Powers of Cotangent
- King's Rule with t-Formula
- Reduction Formula - Logarithmic Powers
- Applications - Particle Dynamics
- Power Reduction Method
Part 2 (48 problems)
Easy (15 problems)
This tier contains 15 problems aimed at foundational fluency. Representative work includes "Basic u-substitution" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Basic u-substitution
- Basic substitution with square root
- Reverse chain rule - exponential
- Standard form - arctan
- Basic trig substitution
- Simple integration by parts
- Reverse chain rule - logarithm
- Basic trig integral
- Substitution with definite integral
- Standard form - arcsin
- Basic partial fractions
- Integration by parts - polynomial times trig
- Reverse chain rule - power
- Even function property
- Completing the square for arctan
Hard (17 problems)
This tier contains 17 problems aimed at extension and synthesis. Representative work includes "Trig substitution for x^2 + a^2" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Trig substitution for x^2 + a^2
- Reduction formula with induction
- Advanced partial fractions
- The Beta Function Level 4: The Symmetry Trap
- Volume with washer method
- Substitution transformation proof
- Integration by parts three times (cyclic)
- Trig substitution for a^2 - x^2 with arcsin
- King's property with complex denominator
- Definite integral with series expansion
- Partial fractions with irreducible quadratic
- Integration by parts - product of ln
- Mechanics - Simple harmonic motion with integration
- Reduction formula application
- Volume with shell method
- Advanced definite integral with properties
- Feynman’s Favorite Trick Solving the Dirichlet Integral
Medium (16 problems)
This tier contains 16 problems aimed at exam-standard reasoning. Representative work includes "Integration by parts twice" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.
- Integration by parts twice
- Partial fractions with repeated factor
- Substitution + completing square
- King's property with trig
- Trig substitution for a^2 - x^2
- Reduction formula - powers of sin
- Partial fractions with quadratic
- Integration by parts - ln
- Definite integral with symmetry
- Trig identity + substitution
- Volume of revolution
- t-formula application
- Particle motion with integration
- Substitution creating ln + arctan
- Complex numbers method
- The Beta Function of the Third Degree
Common patterns across the booklet
- Complex numbers: 2 problems — e.g. "King's property with complex denominator"
- Integration: 40 problems — e.g. "Integration by Parts"
- Vectors & geometry: 1 problem — e.g. "Applications - Inclined Plane Dynamics"
- Mechanics: 1 problem — e.g. "Applications - Particle Dynamics"
- Polynomials: 1 problem — e.g. "Partial fractions with repeated factor"
- Trigonometry: 4 problems — e.g. "Standard form - arctan"
Standout and less-seen problem types
These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:
- Advanced Reduction Formula with Induction: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Reduction Formula with Factorial Series: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Substitution Proof: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Applications - Inclined Plane Dynamics: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Volumes of Revolution with Ratio: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
- Trig substitution for x^2 + a^2: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Working through a large booklet
- Review fundamentals first
- Part 1 without solutions on first pass
- Part 2: try, then upside-down hint
- Practise from memory; use appendices for technique lookup
Timed practice: allow about one minute per mark; write legible structure even when practising alone. Error log: tag mistakes as concept, algebra, or misreading. Rotation: do not camp on advanced tier if basics still slip.
Open HSC Integrals and work steadily — 63 problems is a marathon, not a sprint.
Key appendices
Formula sheet — Standard integrals and rules. Use for quick reference or enrichment beyond routine exam questions.
Index by technique — Find problems by method. Use for quick reference or enrichment beyond routine exam questions.
Common substitutions guide — When to use -sub, trig sub, t-formula. Use for quick reference or enrichment beyond routine exam questions.
IBP decision tree — Choose and systematically. Use for quick reference or enrichment beyond routine exam questions.
Rigorous foundations — Riemann sum enrichment. Use for quick reference or enrichment beyond routine exam questions.
IBP patterns — Polynomial × exponential/trig templates. Use for quick reference or enrichment beyond routine exam questions.
For HSC preparation, prioritise the first one or two appendices; later entries reward curious students but are not required for standard papers.
Key conclusion
Integration is a cornerstone Extension 2 technique. Mastery requires recognising patterns, choosing methods confidently, and extensive practice — use the appendices as quick references.
The booklet's closing section reinforces these habits:
- Review fundamentals first
- Part 1 without solutions on first pass
- Part 2: try, then upside-down hint
- Practise from memory; use appendices for technique lookup
Revisit HSC Integrals in the fortnight before trials and redo problems you missed on first pass.
How to study with this booklet
Week 1 fundamentals; weeks 2–4 one technique per week using Appendix index; week 5 volumes; week 6 mixed past-paper integrals
General principles:
- Closed-book first: attempt without notes, then check fundamentals.
- Error log: record concept vs algebra vs reading errors.
- Spaced repetition: redo missed questions after 3 and 7 days.
- Past papers last: fix weak topics here, then sit full papers timed.
Mini-FAQ
Who is the HSC Integrals booklet for?
Extension 2 students aiming to master integration and advanced problem-solving studying Mathematics Extension 1 and Extension 2 under the NSW HSC.
Should I read solutions before attempting problems?
Attempt Part 1 first. Use Part 2 hints only after a genuine try or partial working.
Where can I read the booklet online?
Open HSC Integrals on Vu's Maths Hub — free, no account required.
How many problems are in the booklet?
Roughly 63 practice problems across 82 pages, each with worked solutions.
Is this aligned with NESA?
Topics match Mathematics Extension 1 and Extension 2 outcomes for substitution, integration by parts, partial fractions, reduction formulae, volumes of solids, and trigonometric substitutions. Confirm scope with your teacher and current NESA documentation.
Common mistakes to avoid
- Wrong IBP choice — use the decision tree in Appendix D
- Forgetting to change limits after substitution
- Partial fraction cover-up errors with repeated factors
- Volume setup: wrong axis of rotation or radius function
- Rushing to advanced tiers before basic fluency — build foundations first.
Practice on Vu's Maths Hub
Open the free HSC Integrals on Vu's Maths Hub — 63 problems with full worked solutions.
Related resources:
- How to use Vu's Maths Hub — Integration applications guide
- HSC Mechanics — Integrate acceleration to velocity
More on Vu's Maths Hub
All booklets are free for personal and school use under the CC BY 4.0 licence.
Related resources:
- HSC Trigonometry — Trig for substitutions
- HSC Differential Equations — Integrate to solve DEs
Substitution — exam context
In NSW Mathematics Extension 1 and Extension 2 examinations, substitution routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Integrals booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Integrals and locate items that stress substitution; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Integration by parts — exam context
In NSW Mathematics Extension 1 and Extension 2 examinations, integration by parts routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Integrals booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Integrals and locate items that stress integration by parts; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Partial fractions — exam context
In NSW Mathematics Extension 1 and Extension 2 examinations, partial fractions routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Integrals booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Integrals and locate items that stress partial fractions; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Reduction formulae — exam context
In NSW Mathematics Extension 1 and Extension 2 examinations, reduction formulae routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Integrals booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Integrals and locate items that stress reduction formulae; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Volumes of solids — exam context
In NSW Mathematics Extension 1 and Extension 2 examinations, volumes of solids routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Integrals booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Integrals and locate items that stress volumes of solids; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Definite integral properties — exam context
In NSW Mathematics Extension 1 and Extension 2 examinations, definite integral properties routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Integrals booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Integrals and locate items that stress definite integral properties; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Trigonometric substitutions — exam context
In NSW Mathematics Extension 1 and Extension 2 examinations, trigonometric substitutions routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Integrals booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Integrals and locate items that stress trigonometric substitutions; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.
Syllabus alignment
This booklet supports Mathematics Extension 1 and Extension 2 under the NESA syllabus. It supplements school instruction with 82 pages of extra exam-style practice — not a replacement for class teaching.
Additional exam advice
When sitting Mathematics Extension 1 and Extension 2 exams, allocate time proportional to marks. Practise concise justification in HSC Integrals — NSW markers reward clear communication. Reread the booklet conclusion the night before for a habit checklist.
Why Vu's Maths Hub
Vu's Maths Hub hosts every HSC booklet in a continuous, mobile-friendly viewer — zoom, search, no download required. Maintained by Vu Hung Nguyen; CC BY 4.0 for personal and school use.
More on Reduction formulae
Return to HSC Integrals and filter mentally for reduction formulae. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Volumes of solids
Return to HSC Integrals and filter mentally for volumes of solids. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
More on Definite integral properties
Return to HSC Integrals and filter mentally for definite integral properties. Strong students redo one problem from each tier without notes, then teach the method aloud — if you cannot explain why each step is valid, the fundamentals section needs another pass.
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