Calculus | HSC Extension 2

Assessment evidence

What these examples cover

These examples show assessable skills, not an official HSC topic weighting or prediction. The 2024 NSW syllabuses have their first HSC examination in 2027; students sitting the 2026 HSC continue under the applicable 2017 syllabus.

Teacher check

Teacher verification note: check the mathematics, notation, permitted technology and syllabus fit for your cohort before classroom or assessment use.

3 deterministic examples

Calculus questions with worked solutions

Each question is generated from an existing curriq QDSL archetype at a fixed seed. Marks and catalog content mappings travel with the generated question.

Worked example 1

2 marksext2-calculus-1

Integration by parts Find

integral x sin(2x) dif x

Part a · 2 marks

Evaluate the indefinite integral, omitting the arbitrary constant.

Worked solution

Answer: (((-2 x) cos((2 x))) + sin((2 x))) / 4

Integration By Parts: Take $u=x$ and $dv=sin(2x) dif x$ .
Antiderivative: $(-2x cos(2x)+sin(2x))/(2^2)+C$ .

Catalog reference: ext2_nsbh_2025_q11b_integration_by_parts · seed 41001

Worked example 2

3 marksext2-calculus-1

A symmetric inverse-tangent integral Evaluate

integral_ -2^ 10 6/(x^2-8x+52) dif x

Part a · 3 marks

Give the exact value.

Worked solution

Answer: pi / 2

Complete Square: $x^2-4{{s}}x+13{{s}}^2=(x-2{{s}})^2+9{{s}}^2$ .
Inverse Tangent Substitution: $u=(x-2{{s}})/(3{{s}})$ changes the bounds to -1 and 1.
Evaluate Bounds: $atan(1)-atan(-1)=pi/2$ .

Catalog reference: ext2_cssa25_symmetric_inverse_tangent_integral · seed 41001

Worked example 3

3 marksext2-calculus-1

Tangent half-angle substitution Evaluate

integral_0^(pi/(22)) (1)/(1+cos(2x)) dif x

Part a · 3 marks

Evaluate the definite integral.

Worked solution

Answer: 1/2

Tangent Half Angle: Let $u=tan(2x/2)$ ; the limits become 0 and 1.
Simplify Integrand: The transformed integrand is constant $1/2$ .
Evaluate Bounds: $1/2$ .

Catalog reference: ext2_morris_2025_q11c_tangent_half_angle · seed 41001

Content mapping and common errors

Calculus content identifiers

MEX-C1

Further Integration

Integrate using trigonometric substitution, partial fractions with irreducible quadratics, and integration by parts.

Common student mistakes

·Selecting the wrong trigonometric substitution for the integrand form
·Errors in setting up partial fractions for irreducible quadratics
·Forgetting to back-substitute to the original variable

MEX-C2

Applications of Integration

Find volumes of solids of revolution; apply arc length and surface area integrals.

Common student mistakes

·Forgetting to square the function when computing a volume of revolution
·Incorrect limits of integration when the curve is rotated about the y-axis

Source and methodology

Examples are deterministic curriq catalog generations from existing QDSL archetypes at the published fixed seed. The generator copies only student-facing prompts, mark allocations, topic mappings, display tables or options, checked answers and marking-path results into this resource.

Review the official NSW Curriculum source ↗

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Marking

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What these examples cover

Calculus questions with worked solutions

Worked example 1

Worked example 2

Worked example 3

Calculus content identifiers

Further Integration

Applications of Integration

Source and methodology

Other Extension 2 topics

From the blog

Generate a Calculus worksheet in 30 seconds