Remembering a friend: Azhaghu Roopesh M

Calculus Level 5

For \(|a|>\dfrac{1}{2}\), \[\int \dfrac{dx}{(\sin x + a \sec x)^2} =\] \[\dfrac{1}{\alpha a^2 - 1}\times \dfrac{\cos 2x}{\beta a +\sin 2x} + \dfrac{4a}{(\alpha a^2-1)^{\frac{3}{2}}}tan^{-1} \left(\dfrac{2a \tan x +1}{\sqrt{\alpha a^2-1}}\right) - \dfrac{1}{\beta a+\sin 2x} + C\]\[\]If \(p+q+r=0\) and \(p^2+q^2+r^2 = \alpha-\beta-1\), then the maximum value of\[p^{333}+q^{333}+r^{333} \] is equal to \(\dfrac{j^k - j}{l^m\sqrt{l}}\).\[\] Find \(j+k+l+m\).

This is part of Ordered Disorder.
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