Level 3 Integration Walkthrough
2019 NCEA Level 3 Integration Question 2(c)
2019 Paper
Question
Find \(k\) such that \(\int_3^k\frac{8}{2x-5}\,dx=10\).
You must use calculus and show the results of any integration needed to solve the problem.
First walkthrough idea
Focus to try first
The denominator is linear, so the antiderivative is logarithmic; then solve the resulting logarithmic equation for \(k\).
Step 1
Integrate the reciprocal
Because \(\frac{d}{dx}(2x-5)=2\), the factor \(8\) becomes \(4\) in front of the logarithm.
Show the first step’s working
Walkthrough overview
What this question practises
This 2019 walkthrough is part of AS91579 — Apply integration methods in solving problems.
Method: Solving for an upper limit from a logarithmic integral.
This is Question 2(c) from the 2019 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.
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Learning summary
Review the method, not only the answer
This walkthrough helps you practise solving for an upper limit from a logarithmic integral. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Keep logarithm domain restrictions and any inner-function factor visible throughout the working.