Level 3 Integration Walkthrough

2018 NCEA Level 3 Integration Question 1(c)

2018 Paper

← Back to paper

Question

\[ \text{Find }\int_6^8\frac{2x-7}{x-5}\,dx. \]

You must use calculus and show the results of any integration needed to solve the problem.

First walkthrough idea

Focus to try first

Rewrite the numerator in terms of \(x-5\) so the fraction splits into familiar pieces.

Step 1

Split the fraction

Express \(2x-7\) as a multiple of \(x-5\) plus a remainder.

Show the first step’s working
\[ 2x-7=2(x-5)+3, \qquad \frac{2x-7}{x-5}=2+\frac3{x-5}. \]

Walkthrough overview

What this question practises

This 2018 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Algebraic division before logarithmic integration.

This is Question 1(c) from the 2018 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.

Page updated .

Learning summary

Review the method, not only the answer

This walkthrough helps you practise algebraic division before logarithmic integration. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check the antiderivative by differentiating it, and handle constants and bounds explicitly.

Continue practising

← Question 1(b) Question 1(d) →