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Calculus Level 4

\[\large 2\int^{\frac{\pi}{2}}_{\frac{\pi}{4}}\cot^{2017}(x) \ dx = (-1)^{a}\left(\sum^{a}_{n=1}\frac{(-1)^{n}}{n}+\ln(b)\right)\]

If \(a\) and \(b\) are positive integers satisfying the equation above, find \(a+b\).

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