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Calculus Level 4

\[ \large \lim_{x\to\infty} \dfrac{x^2 f''(x)}{f(x) } \lim_{t \to0} (1+\sin t)^{1 - \cos t} = n^3 - 5n^2 + 2n + 10 \]

Let \(f\) be a polynomial of degree \(n\) satisfying the equation above. Given that \(f'(x) \) has only one real root and \(f(1) = f(-1) \).

If there exists a real \(k\) satisfying \(f'(k) = 0 \), find \( \displaystyle \int_{2k+1}^{k^2-k+1} f(x) \, dx \).

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