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Calculus Level 5

\[\large \int_0^{2016} H_x^{(2)} \, dx \]

Let \({ H }_{ n }^{ (2) }\) be the generalized harmonic number of order \(n\) of 2, \( \displaystyle H_n ^{(2)} = \sum_{k=1}^n \dfrac 1{k^2} \).

If the integral above is equal to \( a\zeta(b) - H_c \), where \(a,b\) and \(c\) are positive integers, find \(a+b+c\).

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