Level 3 Integration Walkthrough
2025 NCEA Level 3 Integration Question 1(c)
2025 Paper
Question
It is given that \[ \int_{1}^{2}\left(a-\frac{6k}{x^2}\right)\,dx=3 \quad \text{and} \quad \int_{1}^{k}6x\,dx=a, \] where \(a\) and \(k\) are constants.
Determine the possible value(s) of \(k\).
You must use calculus and show the results of any integration needed to solve the problem.
First walkthrough idea
Hint to try first
Use the first integral to write \(a\) in terms of \(k\).
Step 1
Integrate the first expression
The constant term integrates to \(ax\), and the \(x^{-2}\) term becomes a positive \(x^{-1}\) term.
Show the first step’s working
The constant term integrates to \(ax\), and the \(x^{-2}\) term becomes a positive \(x^{-1}\) term.
Key result
\[ F(x)=ax+\frac{6k}{x} \]Walkthrough overview
What this question practises
This 2025 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Linking two definite integrals to solve for constants.
This is Question 1(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 linking two definite integrals to solve for constants. 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.