Limits + Integration + Summation!

Calculus Level 5

We denote the two functions $f(x) = \lfloor x \rfloor - x$ and $\displaystyle g(x) = \lim_{n\to\infty} \frac{(f(x))^{4n}-1}{(f(x))^{4n}+1}$ .

Find the value of $\displaystyle \int \sum_{r=1}^{2014} (g(x))^r \, dx$ .