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

Open a question for hints and a progressive worked solution, or use the standard and paper links to return to its assessment context.

NCEA Level 3 · AS91579

2025 Integration

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) · Estimating cross-sectional area with the Estimating cross-sectional area with the trapezium rule. Question 3(b) · Logarithms Logarithmic antiderivatives with a linear inside.

NCEA Level 3 · AS91579

2024 Integration

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) · Expanding a square before integrating Expanding a square before integrating term by term. Question 2(d) · Trig identities in a shaded sin to Using trig identities in a shaded \(\sin^2x\) area. Question 2(e) · Integration by Substitution Logarithmic substitution in a spherical mass model. Question 3(c) · Decomposing a rational integrand before Decomposing a rational integrand before solving for a parameter.

NCEA Level 3 · AS91579

2023 Integration

Question 1(a) · Splitting an integral into three familiar Splitting an integral into three familiar antiderivatives. Question 2(c) · A rational integrand before solving for k Rewriting a rational integrand before solving for \(k\). Question 3(a) · Simpson’s Rule with measured garden Using Simpson’s Rule with measured garden ordinates. Question 3(b) · Radicals and Surds Simplifying radicals before integrating term by term. Question 3(c) · Product-to-sum in a shaded trigonometric 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

Question 1(a) · Logarithms Splitting an integral into logarithm and trig parts. Question 1(c) · Sin to the power 2(2x) before integrating Rewriting \(\sin^2(2x)\) before integrating exactly. Question 3(a) · Reverse Chain Rule Integration Reverse chain rule on a linear power. Question 3(b) · The Trapezium Rule from tabulated values Applying the Trapezium Rule from tabulated values. Question 3(c) · A rational integrand before evaluating it Simplifying a rational integrand before evaluating it.

NCEA Level 3 · AS91579

2021 Integration

Question 2(c) · Product-to-sum before integrating a trig Using product-to-sum before integrating a trig product. Question 3(a) · Expanding (x+square root of x) to the Expanding \((x+\sqrt{x})^2\) before integrating. Question 3(b) · The Trapezium Rule with a width of Applying the Trapezium Rule with a width of \(0.25\).

NCEA Level 3 · AS91579

2020 Integration

Question 1(a) · Logarithms Power, constant, and logarithmic antiderivatives. Question 1(c) · A rational integrand before evaluating it Rewriting a rational integrand before evaluating it. Question 2(b) · Simpson’s Rule to tabulated values Applying Simpson’s Rule to tabulated values. Question 2(c) · Reverse chain-rule integration of a 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

Question 1(b) · The Trapezium Rule to tabulated values Applying the Trapezium Rule to tabulated values. Question 1(c) · Product-to-sum before evaluating a trig 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) · A rational integrand before evaluating it Rewriting a rational integrand before evaluating it.

NCEA Level 3 · AS91579

2018 Integration

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

NCEA Level 3 · AS91579

2017 Integration

Question 1(e) · A mean value using a 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) · The Trapezium Rule to courtyard Applying the Trapezium Rule to courtyard measurements. Question 3(c) · A rational function to find the exact 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 19 July 2026.