Trigonometrica Hyperbolica

Calculus Level 5

\[\large{ \begin{align} \text{X} & = \int_{0}^{\infty} \dfrac{\cos \left(\frac{\pi x}{2016} \right)}{\cosh (\pi x)} \sin (\pi x^2) \, \mathrm{d}x \\ \\ \text{Y} & = \int_{0}^{\infty} \dfrac{\cos \left(\frac{\pi x}{2016} \right)}{\cosh (\pi x)} \cos (\pi x^2) \, \mathrm{d}x \end{align}} \]

The value of \( \dfrac {\text{X}}{\text{Y}} \) can be expressed as \( \tan \left[ \dfrac \pi {\text{M}} \left( 1 - \dfrac1{\text{N}^2} \right) \right] \), where \(\text{M}\) and \(\text{N}\) are positive integers. Find \(\text{M}\times \text{N}\).

Notation: \(\cosh(x) = \dfrac{e^x + e^{-x}}2 \) denotes hyperbolic cosine function.

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