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We denote the two functions f(x)=⌊x⌋−x f(x) = \lfloor x \rfloor - x f(x)=⌊x⌋−x and g(x)=limn→∞(f(x))4n−1(f(x))4n+1\displaystyle g(x) = \lim_{n\to\infty} \frac{(f(x))^{4n}-1}{(f(x))^{4n}+1} g(x)=n→∞lim(f(x))4n+1(f(x))4n−1.
Find the value of ∫∑r=12014(g(x))r dx \displaystyle \int \sum_{r=1}^{2014} (g(x))^r \, dx ∫r=1∑2014(g(x))rdx.
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