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 ∫v1dvfor v>0v > 0v>0. Each economist proposes a different general form for the result:
Amara: ∫1v dv=lnv\int \frac{1}{v} \, dv = \ln v∫v1dv=lnv
Biruk: ∫1v dv=Alnv\int \frac{1}{v} \, dv = A \ln v∫v1dv=Alnv
Chen: ∫1v dv=ln(kv)\int \frac{1}{v} \, dv = \ln(kv)∫v1dv=ln(kv)
Dalia: ∫1v dv=lnv+C\int \frac{1}{v} \, dv = \ln v + C∫v1dv=lnv+C
(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.
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.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions 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, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
