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Question 709

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]
Markscheme

Integration Questions

  1. A Level
  2. /Maths
  3. /Integration

864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.

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