Level 3 Integration Walkthrough

2025 NCEA Level 3 Integration Question 2(b)

2025 Paper

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Question

The rate of change of a particular function, \(p\), is given by \[ \frac{dp}{dt}=5\cos(4t). \]

Find the function, given that \(p=8\) when \(t=\frac{\pi}{24}\).

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

First walkthrough idea

Hint to try first

Integrate \(5\cos(4t)\) first to get the general function.

Step 1

Integrate the rate

Reverse the chain rule by dividing by the inside coefficient \(4\).

Show the first step’s working

Reverse the chain rule by dividing by the inside coefficient \(4\).

Key result

\[ p=\frac{5}{4}\sin(4t)+C \]

Walkthrough overview

What this question practises

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

Method: Integrating a rate function and fitting the constant.

This is Question 2(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 rate function 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) →