calculus
posted by andy .
If f(x)=∫ (from x to x) cos(t)/(1+e^t) dt and x=asin(77/85), then the absolute value of f′(x)−f(x) can be expressed as a/b, where a and b are coprime positive integers. What is the value of a, and what is the value of b?

The integral is from x to x, so you can replace the integrad by its even part. Since cos(t) is an even function, you can replace the factor
1/[1+exp(t)] by its even part:
1/[1+exp(t)] + 1/[1+exp(t)] =
1/[1+exp(t)] + exp(t)/[1 + exp(t))] = 1
So, the even part of 1/[1+exp(t)] equals 1/2. 
so what is the answer to the question?

I have reduced that to a trivial problem. The whole point of this problem is to find out how to solve it, not to get the correct answer, although I know that this is part of the Brilliant competition and the correct answer does matter here, which simply shows that this is a stupid competition.
If this were a normal advanced high school or first year university level homework problem, the student who writes down the correct solution method, but gets the answer wrong due to some minor mistake will get a top grade, while the student who gets the answer correct but whose explanation is not complete can forget about getting a top grade, he may even get a failing grade. 
However, I am still confused as to how I would proceed.

I would still like help as to how I would proceed. That is why I still ask for the answer.

Brianna bought her car for $15,640. It is expected to depreciate an average of 15% each year during the first 7 years. What will the approximate value of her car be in 7 years?