Level 3 Integration Walkthrough

2019 NCEA Level 3 Integration Question 3(c)

2019 Paper

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Question

Question 3(c) original exam prompt; text transcription follows

First walkthrough idea

Focus to try first

Rewrite \(\frac{x}{x+1}\) as \(1-\frac{1}{x+1}\), then integrate term by term.

Step 1

Rewrite the fraction

Make the numerator look like the denominator minus \(1\).

Show the first step’s working
\[ \frac{x}{x+1} = \frac{x+1-1}{x+1} \] \[ = \frac{x+1}{x+1}-\frac{1}{x+1} = 1-\frac{1}{x+1} \]

Walkthrough overview

What this question practises

This 2019 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Rewriting a rational integrand before evaluating it.

This is Question 3(c) from the 2019 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 3(b) Question 3(d) →