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1 Unit 1: Pure Mathematics A

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Question 334

Four economists, Amara, Biruk, Chen, and Dalia, are attempting to determine the indefinite integral representing a total utility function:

∫1v dv \int \frac{1}{v} \, dv ∫v1​dv

for v>0v > 0v>0. Each economist proposes a different general form for the result:

Amara: ∫1v dv=ln⁡v\int \frac{1}{v} \, dv = \ln v∫v1​dv=lnv

Biruk: ∫1v dv=Aln⁡v\int \frac{1}{v} \, dv = A \ln v∫v1​dv=Alnv

Chen: ∫1v dv=ln⁡(kv)\int \frac{1}{v} \, dv = \ln(kv)∫v1​dv=ln(kv)

Dalia: ∫1v dv=ln⁡v+C\int \frac{1}{v} \, dv = \ln v + C∫v1​dv=lnv+C

a.

(i) Explain why Amara's result is incomplete.

(ii) Explain why Biruk's answer is incorrect as a general anti-derivative of 1v\frac{1}{v}v1​.

[2]
b.

Using the properties of logarithms, demonstrate why Chen and Dalia's forms can be considered equivalent for certain values of the constants kkk and CCC.

[2]
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1 Unit 1: Pure Mathematics A Questions

  1. A Level
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