Question

Rate of Change of Revenue The rate of change of revenue (in dollars per calculator) from the sale of x calculators is

R'(x)=(x+1)ln(x+1).
Find the total revenue from the sale of the first 12 calculators. (Hint: In this exercise, it simplifies matters to write an anti-derivative of x+1 as (x+1)2/2 rather than x^2/2+x).

Answer 1

Assuming no revenue from no calculators being sold, so that
`R(0)=0`, we have

`R(12)=R(0)+\displaystyle\int_0^(12)R'(x)\,\mathrm dx`

`R(12)=\displaystyle\int_0^(12)(x+1)\ln(x+1)\,\mathrm dx`

Integrate by parts, taking

`u=\ln(x+1)\implies\mathrm du=(\mathrm dx)/(x+1)`

`\mathrm dv=x+1\,\mathrm dx\implies v=\frac{(x+1)^2}2`

Then

`R(12)=\displaystyle\frac{(x+1)^2}2\ln(x+1)\bigg|_0^(12)-\frac12\int_0^(12)x+1\,\mathrm dx`

`R(12)=\frac{169}2\ln13-\frac{(x+1)^2}4\bigg|_0^(12)`

`R(12)=\frac{169}2\ln13-\left(\frac{169}4-\frac14\right)\approx174.74`