Level 3 Integration Walkthrough

2025 NCEA Level 3 Integration Question 3(b)

2025 Paper

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Question

Find \[ \int_{1}^{k}\frac{10}{2x-1}\,dx, \] giving your answer in terms of \(k\), where \(k\) is a constant and \(k>1\).

First walkthrough idea

Hint to try first

This is a \(\frac{1}{\text{linear}}\) integral, so a logarithm should appear.

Step 1

Identify the antiderivative

The reverse chain rule turns the coefficient \(10\) into \(5\).

Show the first step’s working

The reverse chain rule turns the coefficient \(10\) into \(5\).

Key result

\[ 5\ln|2x-1|+C \]

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Logarithmic antiderivatives with a linear inside.

This is Question 3(b) from the 2025 NCEA Level 3 Integration paper for AS91579 — Apply integration methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91579.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise logarithmic antiderivatives with a linear inside. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Keep logarithm domain restrictions and any inner-function factor visible throughout the working.

Continue practising

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