Level 3 Integration Walkthrough

2022 NCEA Level 3 Integration Question 2(c)

2022 Paper

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Question

Consider the differential equation

\[ \frac{dy}{dx}=\frac{1}{3y^2(x-1)}, \qquad x>1. \]

Given that \(y=-1\) when \(x=2\), find the value(s) of \(x\) which give a \(y\)-value of \(1\).

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

First walkthrough idea

Focus to try first

Separate variables first, then use the condition to find the constant before substituting the target \(y\)-value.

Step 1

Separate the variables

Move the \(y\)-terms to the left and the \(x\)-terms to the right before integrating.

Show the first step’s working
\[ \frac{dy}{dx}=\frac{1}{3y^2(x-1)} \] \[ 3y^2\,dy=\frac{1}{x-1}\,dx \]

Walkthrough overview

What this question practises

This 2022 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Separating variables and using a condition in a log model.

This is Question 2(c) from the 2022 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 separating variables and using a condition in a log model. 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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