Limits + Integration + Summation!

Calculus Level 5

We denote the two functions f(x)=xx f(x) = \lfloor x \rfloor - x and g(x)=limn(f(x))4n1(f(x))4n+1\displaystyle g(x) = \lim_{n\to\infty} \frac{(f(x))^{4n}-1}{(f(x))^{4n}+1} .

Find the value of r=12014(g(x))rdx \displaystyle \int \sum_{r=1}^{2014} (g(x))^r \, dx .

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