Level 3 Integration Walkthrough

2023 NCEA Level 3 Integration Question 2(b)

2023 Paper

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Question

Solve the differential equation

\[ \frac{dy}{dx}=(4x+1)^{-1/2}, \]

where \(x\ge 0\), given that when \(x=6\), \(y=7.5\).

First walkthrough idea

Focus to try first

Integrate the right-hand side first, then use the point \((6, 7.5)\) to determine the constant.

Step 1

Integrate the derivative

The inside derivative \(4\) has to be accounted for.

Show the first step’s working

The inside derivative \(4\) has to be accounted for.

Key result

\[ \frac{\sqrt{4x+1}}{2}+C \]

Walkthrough overview

What this question practises

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

Method: Integrating a radical differential equation and fitting the constant.

This is Question 2(b) 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 integrating a radical differential equation and fitting the constant. 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 2(a) Question 2(c) →