Definite integral Problem-1

Calculus Level 5

If given that \[\displaystyle \int_{0}^{2\pi} \dfrac{dx}{31+5\cos x+6\sin x}=\dfrac{\pi}{a} , \] find the value of \[\large \left \lceil \frac{a^2}{2} \right \rceil.\]

Clarification: \(\lceil .. \rceil\) denotes ceiling function.

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