How much money did the soda machines generate during the second semester?

[Momentum09]

Homework Statement

In a West Texas school district the school
year began on August 1 and lasted until May
31. On August 1 a Soft Drink company in-
stalled soda machines in the school cafeteria.
It found that after t months the machines
generated income at a rate of
f(t) =
500t/ (5t^2 + 2)
dollars per month. Find the total income,
$Tsec, produced during the second semester
beginning on January 1.

Homework Equations

I know to find the income we first have to find the integral for the function t. I did this and got 50 ln [5x^2 +2].
I set u = 5t^2 +2, so du = 10t dt

The Attempt at a Solution

I used 6 as my upper limit and 1 as my lower limit but I can't get the right answer. Can someone please tell me what is the right limit of integration to use in this case?
Thank you so much for your help!

 

[Defennder]

Your limits are a little off. Why did you use 6? The machine was functioning for 5 months from Aug 1 to Jan 1 the next year. And it's not supposed to be 0 either for the lower limits. That ignores the revenue for the first month.