Integration Techniques NCEA Practice Questions | Calc.nz

Skill overview

What these questions practise

The matched methods explicitly include reverse-chain patterns, substitution, trigonometric identities, algebraic rewriting, logarithmic forms, and the Trapezium or Simpson's Rule.

Relevant standards: NCEA Level 3 Integration — AS91579

This collection contains 49 matching walkthroughs across 9 examination years.

The grouping is based on the method recorded for each walkthrough. Use the linked NZQA standard and paper material for authoritative wording, diagrams, and assessment information.

Before you start

Common mistakes in these methods

Check the derivative of the inner expression and account for any constant factor when reversing a chain-rule pattern.
Transform the full integrand under the substitution, including the differential and any definite-integral limits.
Apply the trigonometric identity before integrating and preserve its constant factors.

Related practice

Integration Techniques questions from different years

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NCEA Level 3 · AS91579 2025 Integration 6 questions

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Question 1(a) — Trigonometric Antidifferentiation Recognising a reverse derivative with a sec-tan pattern. Question 1(d) — Integration and Area Using product-to-sum before integrating a shaded trig area. Question 2(a) — Reverse Chain Rule Integration Reverse chain rule integration of a linear power. Question 2(e) — Integration by Substitution Substitution in a shaded trigonometric area problem. Question 3(a) — Numerical Integration Estimating cross-sectional area with the trapezium rule. Question 3(b) — Logarithms Logarithmic antiderivatives with a linear inside.

NCEA Level 3 · AS91579 2024 Integration 6 questions

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Question 1(a) — Trigonometric Antidifferentiation Recognising a reverse sec-tan derivative pattern. Question 1(d) — Differential Equations Product-to-sum before integrating a differential equation. Question 2(a) — Algebraic Integration Expanding a square before integrating term by term. Question 2(d) — Trigonometric Integration Using trig identities in a shaded sin squared x area. Question 2(e) — Integration by Substitution Logarithmic substitution in a spherical mass model. Question 3(c) — Rational-function Integration Decomposing a rational integrand before solving for a parameter.

NCEA Level 3 · AS91579 2023 Integration 6 questions

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Question 1(a) — Algebraic Integration Splitting an integral into three familiar antiderivatives. Question 2(c) — Rational-function Integration Rewriting a rational integrand before solving for k. Question 3(a) — Numerical Integration Using Simpson’s Rule with measured garden ordinates. Question 3(b) — Radicals and Surds Simplifying radicals before integrating term by term. Question 3(c) — Trigonometric Integration Using product-to-sum in a shaded trigonometric area. Question 3(e) — Differential Equations Factor cancellation in a separable differential equation.

NCEA Level 3 · AS91579 2022 Integration 5 questions

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Question 1(a) — Logarithms Splitting an integral into logarithm and trig parts. Question 1(c) — Sin squared (2x) before integrating exactly Rewriting sin squared (2x) before integrating exactly. Question 3(a) — Reverse Chain Rule Integration Reverse chain rule on a linear power. Question 3(b) — Numerical Integration Applying the Trapezium Rule from tabulated values. Question 3(c) — Rational-function Integration Simplifying a rational integrand before evaluating it.

NCEA Level 3 · AS91579 2021 Integration 3 questions

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Question 2(c) — Trigonometric Integration Using product-to-sum before integrating a trig product. Question 3(a) — Algebraic Integration Expanding (x + square root of x) squared before integrating. Question 3(b) — Numerical Integration Applying the Trapezium Rule with a width of 0.25.

NCEA Level 3 · AS91579 2020 Integration 6 questions

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Question 1(a) — Logarithms Power, constant, and logarithmic antiderivatives. Question 1(c) — Rational-function Integration Rewriting a rational integrand before evaluating it. Question 2(b) — Numerical Integration Applying Simpson’s Rule to tabulated values. Question 2(c) — Chain Rule Reverse chain-rule integration of a square root. Question 3(a) — Trigonometric Antidifferentiation Recognising the reverse sec-tan derivative. Question 3(e) — Integration and Area Product-to-sum in an area between cosine curves.

NCEA Level 3 · AS91579 2019 Integration 5 questions

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Question 1(b) — Numerical Integration Applying the Trapezium Rule to tabulated values. Question 1(c) — Trigonometric Integration Using product-to-sum before evaluating a trig integral. Question 2(a) — Reverse Chain Rule Integration Reversing the chain rule for an exponential term. Question 3(a) — Reverse Chain Rule Integration Reverse chain rule integration of a cubic power. Question 3(c) — Rational-function Integration Rewriting a rational integrand before evaluating it.

NCEA Level 3 · AS91579 2018 Integration 5 questions

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Question 1(c) — Logarithms Algebraic division before logarithmic integration. Question 2(a) — Trigonometric Antidifferentiation Reverse trigonometric derivative patterns. Question 2(c) — Trigonometric Integration A double-angle identity and the first positive trig solution. Question 3(a) — Logarithms Expansion, the power rule, and logarithmic integration. Question 3(b) — Numerical Integration Simpson's Rule and its alternating coefficients.

NCEA Level 3 · AS91579 2017 Integration 7 questions

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Question 1(e) — Trigonometric Integration Finding a mean value using a squared-trigonometric identity. Question 2(a) — Logarithms Logarithmic integration with a linear denominator. Question 2(b) — Reverse Chain Rule Integration Reversing the chain rule for a linear power. Question 2(d) — Integration and Area Product-to-sum before integrating a trigonometric area. Question 3(b) — Numerical Integration Applying the Trapezium Rule to courtyard measurements. Question 3(c) — Rational-function Integration Rewriting a rational function to find the exact courtyard area. Question 3(e) — Integration by Substitution Substitution in a parameterised differential equation.

Keep practising

Antidifferentiation Differential equations

Page updated 9 August 2026.