Hot Integral - 21

Calculus Level 5

\[\displaystyle \int\limits_{-\infty}^{\infty} x(\cos(2\pi nx) - i\sin(2\pi nx))\frac{x+|x|^{-3/4}\sqrt{|x|}}{\sqrt{|x|}} dx\]

Given the above integral can be expressed in the form below :

\[\displaystyle \frac{\delta(n)}{A\pi^B|n|^{C/D}}-\frac{En^F}{G\pi^H|n|^{I/J}} - i(\frac{\sqrt{K+\sqrt{L}}n^M\Gamma(\frac{O}{P})}{Q\pi^{R/S}|n|^{T/U}}))\]

where \(n \in (-\infty,\infty)\) and \(i=\sqrt{-1}\).

Calculate \(A+B+C+D+E+F+G+H+I+J+K+L+M+O+P+Q+R+S+T+U\)

Note : \(\delta(.)\) represents delta function and use \(sgn(x)=\frac{x}{|x|}\).


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