Level 3 Integration Walkthrough

2024 NCEA Level 3 Integration Question 1(d)

2024 Paper

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Question

Consider the differential equation \[ \frac{dy}{dx}=24\cos(3x)\sin(x). \]

Given that \(y=6\) when \(x=\frac{\pi}{3}\), find the value(s) of \(y\) when \(x=\frac{\pi}{2}\).

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

First walkthrough idea

Hint to try first

The product \(\sin(x)\cos(3x)\) suggests the product-to-sum identity \(2\sin A\cos B=\sin(A+B)+\sin(A-B)\).

Step 1

Rewrite the product

Since \(2\sin x\cos 3x=\sin 4x-\sin 2x\), the outside \(24\) becomes \(12\) times that bracket.

Show the first step’s working

Since \(2\sin x\cos 3x=\sin 4x-\sin 2x\), the outside \(24\) becomes \(12\) times that bracket.

Key result

\[ 12(\sin 4x-\sin 2x) \]

Walkthrough overview

What this question practises

This 2024 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Product-to-sum before integrating a differential equation.

This is Question 1(d) from the 2024 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 product-to-sum before integrating a differential equation. 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(c) Question 1(e) →