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1.8 Integration

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Question 111

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

1.8 Integration Questions

  1. A Level
  2. /Maths
  3. /1.8 Integration

438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.

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