Level 3 Integration Walkthrough

2018 NCEA Level 3 Integration Question 1(b)

2018 Paper

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Question

Solve the differential equation

\[ \frac{dy}{dx}=e^{2x}+\frac{1}{x}, \]

given that \(y=2\) when \(x=1\).

First walkthrough idea

Focus to try first

Integrate the gradient first, then use the given point to determine \(C\).

Step 1

Integrate the gradient

Reverse the chain rule for \(e^{2x}\), and use the logarithm rule for \(1/x\).

Show the first step’s working
\[ y=\int\left(e^{2x}+\frac1x\right)\,dx =\frac12e^{2x}+\ln|x|+C. \]

Walkthrough overview

What this question practises

This 2018 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Integrating a differential equation and using an initial condition.

This is Question 1(b) from the 2018 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 differential equation 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) →