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Calculus Level 5

Given that \[I=\displaystyle\int \limits^{\infty }_{0}\left( \dfrac{1-\cos ( \sqrt[3]{42} x) }{x^{2}} \right) ^{2}\, dx= A\pi.\] Find \(A\).

Bonus Find the closed form of \[\displaystyle\int \limits^{\infty }_{0}\left( \dfrac{1-\cos ( \alpha x) }{x^{2}} \right) ^{2}\, dx \]

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