Level 3 Integration Walkthrough

2020 NCEA Level 3 Integration Question 1(c)

2020 Paper

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Question

Question 1(c) original exam prompt; text transcription follows

First walkthrough idea

Focus to try first

Rewrite the numerator as a multiple of \(x-3\) plus a constant, then integrate the reciprocal term logarithmically.

Step 1

Rewrite the rational function

Write \(5x-11=5(x-3)+4\) so that \(x\) no longer appears in the numerator of the fraction.

Show the first step’s working
\[ \frac{5x-11}{x-3} =\frac{5(x-3)+4}{x-3} =5+\frac{4}{x-3} \] \[ \int_4^8\frac{5x-11}{x-3}\,dx =\int_4^8\left(5+\frac{4}{x-3}\right)\,dx \]

Walkthrough overview

What this question practises

This 2020 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Rewriting a rational integrand before evaluating it.

This is Question 1(c) from the 2020 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 evaluating it. 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 1(b) Question 1(d) →