Level 3 Integration Walkthrough
2025 NCEA Level 3 Integration Question 2(c)
2025 Paper
Question
Find the value of the constant \(k\), given that \[ \int_{0}^{k}\frac{1}{\sqrt{4x+1}}\,dx=1. \]
You must use calculus and show the results of any integration needed to solve the problem.
First walkthrough idea
Hint to try first
Treat the integrand as \((4x+1)^{-1/2}\).
Step 1
Find the antiderivative
The reverse chain rule introduces the factor of \(\frac{1}{2}\).
Show the first step’s working
The reverse chain rule introduces the factor of \(\frac{1}{2}\).
Key result
\[ \frac{\sqrt{4x+1}}{2}+C \]Walkthrough overview
What this question practises
This 2025 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Using a definite integral to determine a constant.
This is Question 2(c) 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 using a definite integral to determine a constant. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.