Respuesta :
Answer:
Total revenue = [tex]\frac{169}{2}\ln(13)-\frac{169}{4}[/tex] dollars
Step-by-step explanation:
[tex]R'(x)=(x+1)\ln(x+1)[/tex]
Integrate with respect to [tex]x[/tex].
[tex]R(x)=[/tex] ∫[tex](x+1)\ln(x+1)[/tex] dx
Take [tex]x+1=u[/tex]
Differentiate with respect to [tex]x[/tex]
[tex]dx=du[/tex]
So,
[tex]R(x)=[/tex] ∫[tex]u\ln(u)\,du[/tex]
Use integration by parts: ∫f g dx = f∫ g dx-∫(∫g dx)f' dx
Therefore,
[tex]R(x)=[/tex] [tex]\frac{u^2}{2}\ln(u)[/tex] [tex]-[/tex] ∫[tex]\frac{u^2}{2}\frac{1}{u}\,du[/tex]
= [tex]\frac{u^2}{2}\ln(u)[/tex] [tex]-[/tex] ∫[tex]\frac{u}{2}\,du[/tex]
Put [tex]u=x+1[/tex]
[tex]R(x)= \frac{(x+1)^2}{2}\ln(x+1)-\frac{(x+1)^2}{4}[/tex]
To find total revenue from the sale of the first 12 calculators, put [tex]x=12[/tex]
[tex]R(x)= \frac{(12+1)^2}{2}\ln(12+1)-\frac{(12+1)^2}{4}\\\\= \frac{(13)^2}{2}\ln(13)-\frac{(13)^2}{4}\\\\=\frac{169}{2}\ln(13)-\frac{169}{4}[/tex]
Total revenue = [tex]\frac{169}{2}\ln(13)-\frac{169}{4}[/tex] dollars
