Level 3 Integration Walkthrough

2025 NCEA Level 3 Integration Question 1(b)

2025 Paper

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Question

Solve the differential equation \[ \frac{dy}{dx}=3\sqrt{x}+\frac{2}{\sqrt{x}}, \] given that \(y=10\) when \(x=4\).

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

First walkthrough idea

Hint to try first

Rewrite the derivative in index form before integrating.

Step 1

Integrate to get the general solution

Increase each power by \(1\), divide by the new power, and keep the constant of integration.

Show the first step’s working

Increase each power by \(1\), divide by the new power, and keep the constant of integration.

Key result

\[ y=2x^{3/2}+4x^{1/2}+C \]

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Integrating a derivative and using an initial condition.

This is Question 1(b) from the 2025 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 derivative and using an initial condition. 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(a) Question 1(c) →