Level 3 Integration Walkthrough

2023 NCEA Level 3 Integration Question 2(c)

2023 Paper

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Question

\[ \text{Find the value of }k,\text{ given that } \int_2^k \left(\frac{6x-3}{2x-3}\right)\,dx = 3k. \]

First walkthrough idea

Focus to try first

Rewrite the rational function into a constant plus a logarithmic term before applying the bounds.

Step 1

Rewrite the fraction

Writing \(6x-3=3(2x-3)+6\) makes the integral much easier.

Show the first step’s working

Writing \(6x-3=3(2x-3)+6\) makes the integral much easier.

Key result

\[ \frac{6x-3}{2x-3}=3+\frac{6}{2x-3} \]

Walkthrough overview

What this question practises

This 2023 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Rewriting a rational integrand before solving for \(k\).

This is Question 2(c) from the 2023 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 rewriting a rational integrand before solving for \(k\). Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 2(b) Question 2(d) →