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Calculus Level 5

Let \(\displaystyle f(n) = \int_0^{\pi /2} \dfrac {dx}{1 + n\sin^2 x} \) and \(g(n) = \dfrac{\pi^2}{(f(n))^2} \). Find \( \displaystyle \sum_{i=0}^{100} g(i) \).

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